22email: andrew@bakan.kiev.ua 33institutetext: Håkan Hedenmalm 44institutetext: KTH Royal Institute of Technology, SE–10044 Stockholm, Sweden
44email: haakanh@math.kth.se 55institutetext: Alfonso Montes-Rodríguez 66institutetext: University of Sevilla, 4180 Sevilla, Spain
66email: amontes@us.es 77institutetext: Danylo Radchenko 88institutetext: ETH Zürich, Mathematics Department, 8092 Zürich, Switzerland
88email: danradchenko@gmail.com 99institutetext: Maryna Viazovska 1010institutetext: École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
1010email: viazovska@gmail.com
Hyperbolic Fourier series
Abstract
. In this article we explain the essence of the interrelation described in [PNAS 118, 15 (2021)] on how to write explicit interpolation formula for solutions of the Klein-Gordon equation by using the recent Fourier pair interpolation formula of Viazovska and Radchenko from [Publ Math-Paris 129, 1 (2019)]. By the words of Fourier pair interpolation, we mean the interpolation of a pair of functions of one variable, related to each other by the Fourier transform.
Hedenmalm and Montes-Rodríguez established in 2011 the weak-star completeness in of the sequence , , , , which is referred to as the hyperbolic trigonometric system. We construct explicitly the sequence in which is biorthogonal to this system and show that it is complete in . The construction involves integrals with the kernel studied by Chibrikova in 1956, similar to what was used by Viazovska and Radchenko for the Fourier pair interpolation on the real line. We associate with each its hyperbolic Fourier series
and prove that it converges to in the space of tempered distributions on the real line. The integral transform of given by
is continuous and bounded on , vanishes at infinity, solves the Klein-Gordon equation , and, by the theorem of Hedenmalm and Montes-Rodríguez, it is determined uniquely by its values at . Applied to the above mentioned biorthogonal system, this transform supplies interpolating functions for the Klein-Gordon equation on the characteristics with , , , where is the Kronecker delta symbol. Under additional decay conditions on , these interpolating functions allow us to restore from its values at . The restriction of any smooth solution of the Klein-Gordon equation to a bounded rectangle can be represented in the form of , but this matter will be pursued elsewhere.
Keywords:
Theta functions Elliptic functions Gauss hypergeometric functionMSC:
81Q05 42C30 33C05 33E051 . Introduction
An important unsolved issue in mathematics is the problem of interpolating oscillatory processes. Uniqueness holds a special place in interpolation. For example, when gas fluctuates in a gas storage, it is necessary to select the position of the pressure gauges in order to fully know about the state of the gas at any point in the storage. Unfortunately, nothing but a few numerical methods, based on considering solutions with extreme value of entropy, are known (see, e.g., lio , ger ). There appear to have been no other approaches to this topic until 2011, when the second and the third authors hed considered uniformly bounded solutions of the Klein-Gordon equation in the plane of the form
| (1.1) |
On a compact subset, such solutions can approximate any continuous solution of this equation. They pioneered the study of the discretized Goursat problem which instead of prescribing the values of a solution on the two intersecting characteristics and assumes that these values are known only along the discrete subset of the characteristics consisting of equidistant points , . It was established in hed that the values , , and , , determine the function uniquely. They obtained the following result, which we develop further in Section 9.
In the present paper we explicitly construct a system of functions , , , , which is biorthogonal to the hyperbolic trigonometric system , , , , such that
| (1.4) |
for all , and , where , if , and , if , for arbitrary . We prove that for each its hyperbolic Fourier series
| (1.5) | ||||
converges to in the space of tempered distributions on the real line. We also show in Remark 4.4 that in case or is -periodic and integrable on the period, the series (1.5) coincides with its usual Fourier series of the argument or , respectively. Using (1.12) below (see also Theorem 8.7), we explicitly restore in (1.1) from its values at the points under the condition (1.11). The proof of the assertion that any smooth solution of the Klein-Gordon equation on a bounded rectangle can be represented in the form (1.1) will be supplied elsewhere. Another aspect of the theory which will be developed elsewhere is the series expansion of which is obtained by insertion of the series (1.5) for the function in the integral (1.1).
1.1 . Key proofs.
Given the properties (1.4), it is easy to deduce from Theorem A that for every the series (1.5) converges in and that the biorthogonal system is complete in as well. We indicate briefly the necessary arguments. The tempered test functions form the so-called Schwartz class on the real axis (see (vlad, , p. 74)), so that for all positive integers , and arbitrary integration by part gives
As the integrands here are in , the coefficients
| (1.6) |
satisfy, for any positive integer ,
| (1.7) |
Hence, for all and any positive integer we have
| (1.8) |
as the series converges in the space (see (vlad, , p. 77)). In view of (1.4), we see that for each the function
is in and satisfies
From Theorem A, it follows that and consequently, for any ,
| (1.9) |
where the series converges in the weighted uniform norm , in view of (1.4) and (1.7). Here, , . Since is dense in we deduce the completeness of the system in , as claimed. As for the convergence of the series (1.5), we take an arbitrary and for we write
It follows from (1.4) and (1.8) that, as , converges in the space to an element . For any test function we have
From (1.9) combined with (1.4), (1.7) and the Lebesgue dominated convergence theorem (nat, , p. 161), we obtain, by passing to the limit as ,
.
This tells us that holds in the space . In conclusion, the hyperbolic Fourier series (1.5) converges to in the space of tempered distributions on the real line.
1.2 . Interpolation formula.
We proceed with the interpolation formula for the Klein-Gordon equation. Let be arbitrary and consider the solution of the Klein-Gordon equation given by (1.1). Then the coefficients of in (1.6) are called the conjugate hyperbolic Fourier coefficients and they equal the values of at the points ,
| (1.10) |
By the estimates (1.4) and manipulations similar to those employed in the proof of (1.9) we get that the conjugate hyperbolic Fourier series of in the right-hand side of the equality (1.9) converges to in the weighted uniform norm on provided that
| (1.11) |
If we multiply both sides of the equality (1.9) by and integrate with respect to over the real line, we restore from its values at the points (see Theorem 8.7), by using the identities (1.15) below,
| (1.12) |
Here, are the interpolating functions for the Klein-Gordon equation given by
The biorthogonal system has the symmetry properties
which for arbitrary and lead to the identities,
| (1.15) |
1.3 . Basic result.
Let and
| (1.16) |
be the Gauss hypergeometric function which can be extended to a nonvanishing holomorphic function on the set (see (bh2, , p. 597, (1.19)(b))). In view of the Barnes expansion (bar, , p. 173)(1908) (see also (whi, , p. 299), (2.8)),
for each positive integer , the function is meromorphic in . Moreover, there exists an algebraic polynomial of degree with real coefficients and leading coefficient equal to such that and
| (1.17) |
where denotes the space of all holomorphic functions in a domain . In addition, all the Taylor coefficients of are real (see (rud, , p. 199), (con, , p. 72)). Our main result follows.
Theorem 1.1
. The system given by
| (1.18) | |||
| (1.19) |
Corrollary 1.2
. Suppose . Then the series
represents a tempered distribution on associated with the regular function .
Remark 1.3
. To prove Theorem 1.1 we first write down in (2.50) and (2.51) the integrands of the integrals (1.18) in terms of the Schwarz triangle function. Then using the change of variables we express in (2.52) these integrals through the elliptic modular function which is the inverse to the Schwarz triangle function. This allows us to prove in Theorem 4.3 the fact of biorthogonality indicated in Theorem 1.1.
allows us to write the formulas (1.18) by using only the values of on the interval ,
1.4 . Connection with the Perron-Frobenius-Ruelle operator.
It can be easily seen that for arbitrary in the space there exists the limit
| (1.22) |
and hence the mapping defines the operator , where and denotes the Lebesgue measure on the real line. This operator is known as the Perron-Frobenius-Ruelle operator corresponding to the even Gauss map defined by the formula if and . Here, assigns to each real the unique number in the interval such that is an even integer (see (kes, , p. 81), (ios, , p. 57)). The properties stated in Lemma 4.1(4) and Lemma 4.2(3,4) below tell us that
| (1.25) |
where
| (1.28) |
1.5 . Outline of the paper.
The paper is organized as follows. In Section 2 we list a few facts about the elliptic modular function following an approach suggested by the first and the second authors in bh2 . The polynomials introduced and investigated by the fourth and fifth author in rad have given reasons for us in Section 3 to introduce and explore a sequence of simpler polynomials, from which the polynomials in rad can be obtained by symmetrization. We devote Section 4 to various properties of the biorthogonal system. Two partitions of the upper half-plane are introduced and developed in Section 5. The results of Section 5 are needed in Section 6 to make a constructive description of the analytic extension of the generating function for the biorthogonal system. This extension is similar to what was proposed and implemented by the fourth and the fifth authors in rad . The results of Sections 5 and 6 are presented in the language of the even-integer continued fractions as suggested by the first author. Some estimates of biorthogonal functions are derived in Section 7. Applications to the Klein-Gordon equation of the obtained results are made in Section 8. In Section 9, Theorem A is extended with the purpose to apply it for partial solution of the issues raised in bhm . Section 10 elaborates on the proofs of the main results. Further explanatory notes are supplied in Section A.
1.6 . Notation.
We begin this section by listing the frequently used notations.
For any , the straight line segment from to is denoted by . We extend interval notation on the reals to line segment notation, so that, e.g., for any two points , we write and .
Following the definitions of (sar, , pp. 6, 40), we denote by the real-valued logarithm defined on , and let be the principal branch of the logarithm defined for with . Furthermore, for a simply connected domain , a point , and a function which is zero-free in with , we write for the holomorphic function in such that , , and (see (con, , p. 94)). Then and for each .
As for topology, we denote by (or ), , and the closure, interior, and boundary of a subset , respectively.
Let denote the linear space of all continuous complex-valued functions on . For a non-negative Borel measure on we denote by the standard normed space of integrable with respect to complex-valued Borel functions. For arbitrary non-negative Borel measures , on and , we write if for arbitrary Borel subset of . If here is the Lebesgue measure on the real line and , , then instead of and we write (or ) and , respectively.
For , we denote by the class of all nontangential limits on of functions from the Hardy space (see (ko, , p. 112), (gar, , p. 57))
and .
2 . The Schwarz triangle function and its inverse
We denote , , and
| (2.1) |
Everywhere below we also use the notation for the contour of integration which is extended from to along the open semicircle .
2.1 . The Gauss hypergeometric function.
Euler’s integral representation
of the Gauss hypergeometric function (1.16) gives its analytic extension from to satisfying
| (2.2) |
and together with the Pfaff formula (see (and, , p. 79))
it allows us to find out the boundary values of on the both sides of the cut along (see (olv, , p. 491, 19.7.3; p. 490, 19.5.1)),
Moreover, it also allows us to estimate for as follows
(see (bh1, , p. 35, (A.9e))), and conclude that belongs to the Hardy space for arbitrary . Then the Schwarz integral formula (see (ko, , p. 128))
| (2.3) |
can be applied to restore on from the values of on (see (bh2, , p. 604, (2.12))). The similar quadratic formula
| (2.4) |
also applies, because is in the Hardy space for all . For arbitrary we use the relation
to derive from (2.4) that
from which we conclude that
| (2.5) |
It follows from (2.3) that has property
which reinforces (2.2), and allows us to form the logarithm of it and to obtain the integral representation
| (2.6) |
(see (bh2, , p. 595, (1.15))), where
because
| (2.7) |
(see (bh2, , p. 606, (3.9))) implies
which is valid for , we have (see (bh2, , p. 602, (2.5)))
| (2.9) |
2.2 . The Schwarz triangle function.
Theorem B
. The Schwarz triangle function
| (2.10) |
maps the set one-to-one onto the ideal hyperbolic quadrilateral
| (2.11) |
which will be referred to as Schwarz quadrilateral (see Fig. 1).
of (2.10), imply
| (2.12) |
and that the function on the interval decreases from to ,
| (2.13) |
To make other known properties of obvious, it is necessary to use the integral representation (see (bh2, , p. 608, (3.13)))
which is immediate from (2.6) for every . This representation has the following convenient properties (see Section 10.1.1).
Lemma 2.1
. Denote . The integrands in
| (2.14) |
and in
| (2.15) |
are of constant sign for each . More precisely, if then
for arbitrary .
2.3 . The elliptic modular function .
The function which is the inverse to , i.e.,
| (2.20) |
The modular function extends to a periodic nonvanishing holomorphic function in with period and (see (bh2, , p. 598, (1.29)))
| (2.21) |
where for and (see (bh2, , pp. 612–614, (6.1),(6.7),(6.8))),
Regarding these nonvanishing holomorphic functions in and in , correspondingly (see (bh2, , p. 598, (1.26))), the main relationships between them can be written for arbitrary as follows, by using the principal branch of the square root,
| (2.25) |
(see (bh2, , p. 614, (6.8)), ). In addition, they are called the theta functions and meet the Jacobi identity
| (2.26) |
and that the function on the interval strictly decreases from to ,
| (2.32) |
The known relationships (see (bh2, , p. 598, (1.25); p. 599, (1.32)))
| (2.38) |
for the principal branches of the quadratic and of the fourth roots together with (2.20) and (2.21), show that the Schwarz triangle function is a key that links the theta functions with the Gauss hypergeometric function . This allows each property of to be formulated as a property of theta functions and vice versa.
For instance, by virtue of (2.4), we have for all , and as a consequence, if we take into account (2.16) and (2.38)(b), we get the result that
| (2.39) |
which was established by the first and the second authors in Corollary 1.2 of (bh2, , p. 599). Our second example is the property
| (2.40) |
which follows from (2.5) and the identity
| (2.41) |
(see (bh2, , p. 599; (1.30))), by application to them (2.38), (2.19)(b) and (2.16), as with the help of (2.38), the identity (2.41) can be written in the form
As for conclusions going in the opposite direction, here we can mention that the combination of (2.38)(b), (2.20), (2.21) and the Landen transformation equations
| (2.44) |
(see (law, , p. 18)), gives the quadratic transformation relation (3.1.10) with of (and, , p. 128) for the hypergeometric function and equality (1.21) as well. At the same time, each of the four nontrivial functional relationships for the modular function in the table of (cha, , p. 111) (except for the first and third, considered as trivial) can be written as the corresponding Kummer transformation rule for (see (erd1, , p. 106)). For instance, in (bh1, , p. 33) the Kummer identity (27) of (erd1, , p. 106)) was derived from (3.38). It also follows from (2.44) that
| (2.45) | ||||||
| (2.46) | ||||||
Observe that the principal branch of the square root can be used in (2.45) because and the two inclusions , hold if and only if . A combination of (2.31)(b) and (2.45) gives that
| (2.47) |
The relation
| (2.49) |
(see (bh2, , p. 597, (1.20))) allow us in Section 10.1.2 to show that the integral formulas of Theorem 1.1 for every can be written as follows,
| (2.50) | ||||
| (2.51) |
2.4 . Monotonicity properties of the modular function .
and (2.40) allow us to obtain in Section 10.1.3 the following monotonicity properties of and the estimates of Corollary 2.3 below as well.
Theorem 2.2
. Let and . Then
| (2.56) | ||||
| (2.57) | ||||
| (2.58) | ||||
| (2.59) | ||||
| (2.60) |
Since holds for , the next properties can be derived from Theorem 2.2, (2.41) and (see (2.31)–(2.35)).
Corrollary 2.3
. Suppose , , and put
Then and
| (2.64) |
2.5 . Imaginary part of the Schwarz triangle function.
We note that enjoys the functional relation
| (2.65) |
(see (bh2, , p. 597; (1.21))) and for every there exist (see (bh2, , p. 604)). Moreover, there are relationships between the values of on the two sides of the cut along :
| (2.66) |
(see (bh2, , p. 597; (1.22))), and along the other cut as well,
| (2.67) |
(see (bh2, , p. 597; (1.23))). It was also explained in (bh2, , p. 601; (4.1)) that the Pfaff formula (see (and, , p. 79)) gives that
| (2.68) |
where for we write (see (bh2, , p. 599))
| (2.71) |
Then, by (2.65),
| (2.72) |
i.e., we have
| (2.73) |
Together with (2.68) this shows that the values of on the two sides of the cuts at and at coincide. Hence, we extend the function , initially defined on , to declaring its values to be given by
| (2.76) |
We find that the resulting function is continuous on . Taking the limit in the relation (2.20)(b) as , we obtain from (2.68) and (2.72) that
| (2.79) |
where maps onto itself in a one-to-one fashion, by (2.71). Together with Theorem B this means the following (see (2.1)).
Lemma 2.4
. The modular function maps each of the sets
and
one-to-one onto and each of the set
one-to-one onto . As a consequence, maps each of the sets
| (2.82) |
one-to-one onto .
By using the property as , , which comes from (2.71), we obtain from (2.76) that and as , , which together with the properties
(see (bh2, , p. 609; Lemmas 4.1, 4.2)) means that the function , which is continuous on , can be continuously extended to the point as well, with value , while . As a result, we get the following statement.
Lemma 2.5
. Let be defined as in (2.71). The harmonic function on can be continuously extended to with values on given by (2.76) and . The extended function is positive on and satisfies
| (2.83) |
It follows from Theorem B, (2.79) and (3.39) that for every the equation with has exactly two solutions with equal imaginary parts, by virtue of (2.76), while for every it has a unique solution given by . In fact, Lemma 5.7 below implies the following stronger property, which we obtain later on in Section 10.1.4.
Theorem 2.6
. For every there exists a finite maximum
attained at one or two points, which belong to . There is one such point if and two points if . In particular
| (2.84) |
where is as in Lemma 2.5.
Corrollary 2.7
. For all pairs with , , and , we have .
3 . Schwarz triangle polynomials
Lemma 3.1
. Let and be periodic with period . Then
| (3.1) |
If is such that then .
3.1 . Definition and connection with Faber polynomials.
The notation (2.10) permits us for arbitrary positive integer to write the decomposition (1.17) in the form
| (3.2) |
and we will call the -th Schwarz triangle polynomial.
A special symmetrization of algebraic polynomials occurs when we consider the expression to make the function in (2.52) invariant up to a multiplier with respect to the argument reversal . In view of (2.21), we have , which suggests the following algebraic operations (see Section 10.2.1).
Lemma 3.2
. Let be a positive integer and be an algebraic polynomial of degree with real coefficients. Then there exist two algebraic polynomials of degree with real coefficients such that for every , we have
For each , the symmetrized functions
| (3.3) |
are holomorphic in and -periodic. They were considered for the first time by the fourth and fifth authors rad in the context of Fourier pair interpolation on the real line.
In accordance with (2.21), for and we introduce the notation
| (3.4) |
Then clearly, we have , , . As we substitute for in (3.2) and apply the left-hand side identity in (2.20), we find
| (3.5) |
In view of (2.21), , and hence the right-hand side function in (3.5) is holomorphic on . In view of Lemma 3.1 applied to the left-hand side function in (3.5) we see that . But since with , the composition is holomorphic in a neighborhood of the origin, so that in fact the function of (3.2) meets
and, in view of the uniqueness theorem for analytic functions (see (con, , p. 78)), the relationship (3.5) can be written as
where . According to the definition of Issai Schur (hur, , p. 34), if is the -th Faber polynomial of then , .
3.2 . The generating function.
The generating function of the Schwarz triangle polynomials is calculated in the next lemma.
Lemma 3.3
. We have
| (3.6) |
where the series converges absolutely and the function is continuously extended to in accordance with Lemma 2.5.
Proof of Lemma 3.3. Let . Then there exists such that . We calculate
and in view of (3.2), we have
| (3.7) |
belong to the boundary of . Making the change of variables for in (3.7) and using (2.20), we obtain and
| (3.8) |
where the contour of integration can be replaced by the straight line segment connecting the two points , as by (2.56),
In view of (2.83), Corollary 2.7 tells us that the integrand in (3.8) is a holomorphic function of the variable when , and hence the periodicity of the integrand in (3.8) allows us to use Lemma 3.1 to shift the segment of integration to for any . This gives
| (3.9) |
We conclude that the series
| (3.10) |
3.3 . Asymptotic behavior for the large index.
The condition in (3.6) for the convergence of the series for the generating function of the Schwarz triangle polynomials is sharp. Indeed, it follows from Lemma 5.7 that for every the set is countable and has no limit points in . Then (2.84) implies that for every there exists such that for all satisfying . This allows us to apply the Cauchy formula in (3.9) for the rectangle with vertices , and obtain
| (3.11) |
3.4 . Symmetry property.
The condition for the convergence of the series in (3.6) is at its weakest when as . In the next lemma we explain this fact by showing in (3.13) that , as , and derive in (3.12)(a) an important symmetric property of the Schwarz triangle polynomials.
Lemma 3.4
. The following identities hold
| (3.12) | ||||
| (3.13) | ||||
| (3.14) | ||||
| (3.15) | ||||
| (3.16) |
We supply the proof of Lemma 3.4 in Section 10.2.1. By the symmetric property (3.12) and the change of variables on the right-hand side integral of (2.52), for arbitrary and we obtain from (1.19), (2.21) and (10.4) that
| (3.17) |
where with . Hence for arbitrary and , we obtain
| (3.18) |
where we have used the following notation for the Taylor series (see (3.14))
| (3.19) |
3.5 . Characteristic behavior at the vertices of the Schwarz quadrilateral.
We note that there are only four distinct Möbius transformations (see (con, , p. 47))
The -th Schwarz triangle polynomial can be fully characterized by the behavior of the function from (3.5) at the vertices of the Schwarz quadrilateral. Note that (3.5) for arbitrary and can also be written as
| (3.21) |
where we use the Taylor series notation
and, consequently, , as . It follows from (2.21), from the identities
| (3.23) |
and thus for arbitrary and , we obtain, by (3.19),
| (3.24) | ||||
| (3.25) |
When tends to one of the vertices of , we assume that if and if , which imply that if , while if . Hence it follows from (3.18), (3.21), (3.24) and (3.25) that for every there exists such that
| (3.29) |
and
| (3.33) |
where for every the functions
| (3.36) |
are holomorphic in and periodic with period . We observe that in this notation the formulas (2.52) and (3.17) can be written in the form
| (3.37) |
where the behavior of the both integrands in is completely described by (3.29) and (3.33). In particular, it follows from (3.29)(a) and (3.33)(a) that the both integrals converge absolutely for all . Furthermore, by writing (3.23) in the form (see (cha, , p. 111))
| (3.38) |
The following Liouville-type property holds for the Schwarz quadrilateral , where in the notation (2.11), and
| (3.39) |
Lemma A ((bh2, , p. 597))
The next lemma shows that the function is uniquely determined by much weaker asymptotic conditions at the vertices of the Schwarz quadrilateral than those of (3.29).
Lemma 3.5
. For a given positive integer there exists a unique function in such that for any we have
| (3.44) |
Lemma 3.5 follows easily from Lemma A. Indeed, the conditions (3.44) are satisfied if (3.36)(a) holds, as follows from (3.29). As for the uniqueness, suppose that the function has all the properties of (3.44) (with replaced by ). Then the difference meets the conditions of Lemma A with . By Lemma A, this difference is a constant function, which must equal zero, by property (3.44)(c). This completes the proof of Lemma 3.5.
The symmetrized functions introduced in (3.3) are characterized in a similar manner.
Lemma 3.6
. For every positive integer , there exist unique functions and in the set such that
On a side note, we observe that the functional properties (3.23) imply that for arbitrary and , we have
4 . The biorthogonal sequence
This section is devoted to the study of the functions , , , for , given in Theorem 1.1.
4.1 . Explicit expressions for the biorthogonal sequence.
from which, in view of (1.18), we obtain
| (4.1) |
The main results of this section are the following assertions, the proofs of which are deferred to Sections 10.3.1.
Lemma 4.1
. For every the functions and have the following properties:
Lemma 4.2
. The function has the following properties:
This concludes the verification of (4.2). By applying the change of the variables in (4.2), the relations (4.3) are immediate, since the functions are connected via the relation (1.19). Expressed differently, the following theorem holds, which contains part of the assertions made in Theorem 1.1.
Theorem 4.3
. The system consisting of the functions , , , , is biorthogonal to the hyperbolic trigonometric system , , , .
Remark 4.4
It follows that the hyperbolic Fourier series (1.5) of the -periodic function is the usual Fourier series of on the interval . Next, assume that , . Then, by (1.19) and the above identities for ,
while, as follows from Lemma 4.2(4) and the above equalities,
It follows that the hyperbolic Fourier series (1.5) of equals the usual Fourier series of on the interval expressed in the variable .
4.2 . The generating function for the biorthogonal sequence.
where for arbitrary , in accordance with (3.9), we have
| (4.5) |
In view of Corollary 2.3, we have and for all and . Hence we obtain
for every and . Together with (2.47), this inequality allows us to apply the Lebesgue dominated convergence theorem (see (nat, , p. 161)) in order to derive from (4.4) and (3.37) that
| (4.6) | ||||
| (4.7) |
where both series converge absolutely and uniformly over all .
5 . Partitions of the upper half-plane
In the Poincaré half-plane model of hyperbolic space the (generalized) semicircles , , , , (see (2.1)) are the hyperbolic straight lines connecting ideal points and , or and , respectively. Moreover, given an arbitrary collection of four (three) points () a bounded set is called an ideal hyperbolic quadrilateral (triangle), with four (three) vertices () if is equal to the union of the three (two) lower arches () and the roof (upper arch) (). We will omit the word ”ideal” in the sequel, and note that clearly, there exists a unique bounded closed set satisfying this definition for any given collection of vertices lying on .
Let be the group of Möbius transformations (called modular)
, , , ,
on with the superposition as a group operation (see (cha, , p. 11)). The theta subgroup of is defined as a collection of all with , satisfying either
or
, while the subgroup of as all with .
Definition 5.1
. For , we say that the set (called orbit of with respect to ) generates a partition of if and for all .
In this section, we consider two partitions of generated by and .
Both partitions arise from the need to analyze integral operators with kernel of the form
and integration over . An operator of this type (but with integration over ) was first studied by Chibrikova in 1956 and later reproduced in Gakhov’s monograph
(gah, , p. 513, (52.3)) from 1966 when considering the Riemann boundary value problem in a fundamental domain with respect to a Fuchsian group of linear fractional transformations. The importance of studying such integral operators was highlighted in the
recent work of the fourth and fifth authors rad , where the following crucial observation was made.
Given, e.g., a bounded function ,
if we integrate of with respect to along the semicircle
, we obtain an analytic function in the domain
which can be analytically extended to all of
, while preserving relevant growth control.
The main goal of introducing the above mentioned partitions of the upper half-plane is to obtain explicit formulas for such an analytic extension, see Section 6 below. The results of Sections 5 and 6 are expressed in the language of the even-integer continued fractions which is the suggestion of the first author. This elaborates on the approach of the second and the third authors hed based on the
connection between even-integer continued fractions and the even Gauss map, see, e.g., pio and lop .
5.1 . Even Gauss map in the upper half-plane.
| (5.1) |
The theta subgroup is generated by the Möbius transformations and (see (cha, , p. 112)). If we write, for , , ,
| (5.2) |
Here, we see that
| (5.6) |
for every , and . Hence, if is invariant under the inversion , i.e., if , then its orbit with respect to can be written in the form
| (5.7) |
This suggests the introduction of the following subset of ,
| (5.8) |
Let and denote the usual fractional and integer parts of , respectively, and put (see (bh2, , pp. 599, 600))
| (5.10) |
For arbitrary we define its even integer part as the average of the endpoints of the interval that the point belongs to in the following interval partition of the real axis,
| (5.11) |
and we define its even fractional part to be . As such, they have the properties , , , for each , and
| (5.12) |
Definition 5.2
. We define the complex analogue of the even Gauss map , , , , associated with the even fractional part , as follows
| (5.13) |
For , we have that
| (5.14) |
We note that
| (5.15) |
Estimating from above the modulus of lying on the union of semicircles we obtain from the identity , , that
| (5.20) |
5.2 . The Schwarz partition of the upper half-plane.
We introduce the notation
where is as in (5.1). We associate with an arbitrary -tuple , , and, correspondingly, with each , its ”sign”
| (5.27) |
Depending on the sign of the transformation , , which we apply to the Schwarz quadrilateral , we add to one of the rays ,
| (5.28) |
and denote the resulting image as follows
| (5.29) |
For each the open set is a hyperbolic quadrilateral with four vertices whose position is completely determined by the value of , as follows from the relationships (see Section 10.4.1)
| (5.32) |
Moreover, for arbitrary the properties (5.32) and ()
| (5.33) |
, imply that is a hyperbolic polygon which consists of the three lower arches , , , which do not belong to , and of the roof , which is in (see Figure 3). Obviously, and .
Lemma 5.3
(a) For arbitrary , , we have .
(b) The lower arches
of are the roofs of the respective quadrilaterals
(c) The roofs of and are and , respectively.
(d) The roof of is the lower arch of
(e) holds,
where it is assumed that .
The proof of Lemma 5.3 is supplied in Section 10.4.2. In the sequel, we agree to use the notation
| (5.34) |
As we combine the properties (5.26)(b) and (5.19)(c) together with the definitions (5.29), (5.28) and (5.27), we find for every and that
| (5.37) |
It follows from Lemma 5.3(a) and (5.29) that the sets , , are disjoint subsets of , and their union equals all of .
Lemma 5.4
. Let . We have the partition
| (5.38) |
where for we use the notation .
which we refer to as the Schwarz partition of . The following property is obtained in Section 10.4.3.
Theorem 5.5
. The set generates the Schwarz partition of .
Remark 5.6
. It is known that the fundamental domain (see (cha, , p. 15)) for the subgroup , (elements written below are linear fractional mappings)
| (5.41) |
generated by the Möbius transformations and (see (cha, , p. 111)), can be chosen by any of the four sets written in (2.82), for instance, , where (cp. (cha, , p. 115)). Then
| (5.42) |
forms a partition of the upper half-plane corresponding to the subgroup . But it follows from (5.41), and (5.42) that
| (5.43) |
and therefore the partition (5.42) coincides with the Schwarz partition (5.40) on the set , because
| (5.44) |
these two partitions are different. This difference is essential, since the sets that make up the Schwarz partition are much simpler than the sets of the partition (5.42). Actually, if the partition (5.42) adds to each open hyperbolic quadrilateral its two sides and , which can be two lower arches or a lower arch and a roof, depending on the sign of , then the Schwarz partition adds to such a quadrilateral its own roof , where and (see the note before Lemma 5.3). Hence, the Schwarz partition (5.40) is an easy-to-use modification of the known partition (5.42) associated with the subgroup . Moreover, the choice of the most convenient partition of the set (5.45) is a separate issue, which is no longer related to the structure of , but depends on the mutual arrangement of the sets .
5.3 . Even rational partition of the upper half-plane.
By virtue of Lemma 2 of (cha, , p. 112) and (5.9), the relationships (5.3) and (5.8) can be expressed in the form (the elements of the sets below are linear fractional mappings)
| (5.46) | |||
| (5.50) | |||
| (5.51) |
where no repetitions of linear fractional maps occur in the listing of the sets on the right-hand sides of (5.51) and (5.46). That this is so follows from Lemma 5.3(a) and the fact that neither of the mappings nor can be an element of the rightmost set of (5.46), where the listed four linear fractional maps are all different as the unique entry in the associated matrix with biggest absolute value (denoted by ) occupies four different positions. Consequently, it makes sense to define the ”order” of each element by
| (5.55) |
Moreover, since the semicircle is invariant under the inversion , we realize that
| (5.59) |
and, in the notation ,
| (5.62) |
holds for every (where ). By Lemma 2.4, (5.62) means that is one-to-one on each set of the partition (5.38), with the exception of , where . This has one useful consequence, the proof of which is supplied in Section 10.4.4.
Lemma 5.7
. Let us write and . Then for arbitrary and we have
| (5.63) | ||||
| (5.67) |
where each set is countable and has no limit points in .
Lemma 5.7 can be used to characterize the sets
| (5.68) |
Indeed, the case in (5.63) applied to the semicircle , gives
| (5.69) |
and it follows from and (5.7) that
| (5.70) |
Regarding the two triangular parts into which the open semicircle splits the Schwarz quadrilateral, we introduce the notation (see (5.28))
| (5.71) |
where . Then
| (5.72) |
and for each we can apply to the equalities (5.72) with , to obtain
| (5.75) |
Here and are hyperbolic triangles, the set forms the vertices of , and is open, while is the roof of . At the same time, are also the vertices of , and are the lower arches of not contained in , while the roof of is contained in .
By regrouping the subsets involved, we find that
| (5.76) |
where we use the notation
| (5.79) |
which leads to ()
| (5.80) |
In particular, the set is open and its boundary consists of the roof and the infinitely many lower arches , , which accumulate at the origin, i.e.,
| (5.81) |
where denotes the convex hull of a given subset .
For every the
associated subset has the similar structure, as follows from the relationships (5.80).
To be specific, is open and its boundary consists of the roof and the infinitely many lower arches
, ,
which accumulate at the point . The analogue of (5.81) reads (see (5.34))
| (5.83) |
In particular, is simply connected for each (see (con, , p. 93)).
The rational number , , , , is called even if . It is known (see (lop, , p. 303)) that each nonzero even rational number on can be uniquely represented in the form with some , and conversely. This suggests the introduction of the following notions.
Definition 5.8
.
Given and , the open sets
, and
are called the even rational neighborhoods of
, and the even rational number ,
respectively.
For each given , we add to the subset its corresponding roof , and obtain the roofed subsets (see (5.33)),
| (5.84) |
which we refer to as the even rational partition of . Here, we write and in accordance with (5.69)(b) and (5.76), it follows from (5.85) that (for notation, cf. (5.68))
| (5.86) |
The following property is obtained in Section 10.4.3.
Theorem 5.9
. The set generates the even rational partition (5.85) of .
By intersecting the both sides of (5.85) with the closed unit disk , we get
| (5.87) |
This for arbitrary shows that (cf. (bon, , p. 44))
so that in view of (5.55),
where for this splitting coincides with (5.87). In accordance with (5.55), (5.80) and (5.85), we can associate with each point the corresponding even rational height
| (5.91) |
The results established in (bon, , p. 44) imply the asymptotics
6 . Analytic continuation of the generating function
For arbitrary and let (see (5.55)),
| (6.1) |
and . It can easily be shown from (5.1) that
| (6.4) |
and the identities (5.79)(b) can be written as
| (6.5) |
The formulas (5.26) for arbitrary and take the form
| (6.6) |
Moreover, we observe that (5.58) gives
| (6.7) |
We let be fixed, and consider the following two functions of ,
| (6.8) |
By (5.59), (5.68) and (2.47), the functions and are holomorphic at each point of , and, in view of (4.6) and (4.7),
| (6.9) |
holds for all , and . It follows from (5.69) that is holomorphic -periodic function on the set . The goal of this section is to prove that for every the function can be analytically extended from this set to . To do this, it suffices to show that such an extension is possible from the set to , because then the desired extension on the remaining set can be constructed from the resulting extension by the formula
| (6.13) |
6.1 . Auxiliary lemmas.
In view of Lemma 2.4, for every , , , we introduce the function as the integral (6.8), where is replaced by the oriented contour , which passes from to along the polygonal contour .
Lemma 6.1
. For each and the function of can be analytically extended from to such that the resulting extension satisfies
| (6.16) |
Proof
. By transforming the contour of integration in (6.8) to and using Lemma 2.4, we obtain from the residue theorem (con, , p. 112) that
| (6.19) |
This means that the function , being holomorphic on , coincides on the set with the function , which is holomorphic on . By the uniqueness theorem for analytic functions (see (con, , p. 78)), we find that the function can be analytically extended from to . Moreover, for , the resulting extension equals the expression since holds for all , in view of (6.19). But by (5.79)(a) and (5.86), we see that , and since is simply connected it follows that the latter function is actually holomorphic on . This proves (6.16) and completes the proof of Lemma 6.1.
Lemma 6.2
. Let , , , and the sets , be defined as in (5.75). The function of can be analytically extended from to such that the resulting extension satisfies
| (6.22) |
where the holomorphic at each function is defined as
| (6.25) |
Proof
. We put if is even, while if is odd. Let the function be given as in Lemma 6.1 with . By (5.72), (5.75) and (6.4), we have , and , in view of (5.79)(a). It now follows that is in . By (6.12), we obtain from (6.16) that, for ,
On the other hand, if then
Hence the function is in , where and for all , while
The desired analytical extension of the function from the set to can be easily constructed by combining the results of Lemmas 6.1 and 6.2.
Lemma 6.3
Let , and be given by (6.8). The function extends analytically from the set to such that the resulting extension on the set
| (6.26) |
satisfies
| (6.27) | ||||
| (6.28) | ||||
| (6.29) |
when for arbitrary and . Here, for every , is defined as in (6.25) and
| (6.30) |
Proof. As follows from and (5.69), we have , and hence (6.26) is immediate from (5.76). Since is holomorphic at each point of we conclude from (6.26) that is holomorphic on for every .
What remains to prove is that extends analytically across for each . In the case when , this fact follows from Lemma 6.1 because, in accordance with (6.16), for all . If and then is the roof of and is the lower arch of , by virtue of (5.75). According to (5.79)(a), and , if , while , if . Hence, in this case, the required property follows from Lemma 6.2, because for each , the difference equals the function
6.2 . Main result.
where and . Consequently, for each and , the formula for in Lemma 6.3 can be written in the form
where, for every , the sum has only finitely many nonzero terms, the number of which equals (see (5.91), (6.31) and (6.32))
| (6.33) |
and hence, in view of (6.6), we find that
| (6.36) |
By applying the identity
In view of (6.36), for every , and we can write the function from (6.29) in the form (see (6.25))
| (6.38) |
where and the properties (6.36), (6.37) hold. On the other hand, in the formula (6.29) the values of the function for can be expressed with the help of (6.12) through its values on :
| (6.39) |
because , by virtue of (6.4), and for all . Moreover, the first equality in (6.19) actually holds for all . Indeed, by (5.98)(b) we have , while , as follows from (2.35)(c). It follows that holds for all and . Since is simply connected and , , we can transform the contour of integration in (6.8) to for each to get and hence
| (6.40) |
A similar change of the contour can be made in (6.28) and in (6.27) after application (6.12) to (6.27) with . Combining this with (6.40), (6.39), (6.38), (6.36), (6.37) and Lemma 6.3 gives the main result of this section.
Let
Here the contour passes from to , , and the convergence of both integrals is absolute, in view of (2.47) and (7.2).
Theorem 6.4
For each and the function of the variable can be analytically extended from the set to such that the resulting extension on the set (see (5.86))
| (6.41) |
for , , , and arbitrary , satisfies
| (6.46) |
We observe that by the definition (5.14) of and the identity , , we have for every and thus
| (6.47) |
Since for every we have , , then
| (6.50) |
7 . Evaluation of biorthogonal functions for large index
where the value of is taken from (ber1, , p. 325) and the principal branch of the square root is used. From (3.38) and (2.31)(b) we thus obtain
| (7.2) |
7.1 . Evaluation of the generating function.
Using the identity , , written in (2.41), and the fact that all three values of the variable used in the formula (6.46) for the function belong to , this function can be estimated as
For arbitrary and we obviously have
and hence
| (7.3) |
Since the sets and in (6.46) are the subsets of and for any from these sets we have , (, if ), in view of (6.36), (5.87) and (5.84), we deduce from (7.3) that
As the roof of is also the roof of then, in accordance with Lemma 5.3(e), the property implies that , i.e., and, by (6.50), we obtain for the latter sum the estimate
| (7.4) |
As a result, we obtain the following estimates for the functions from (6.46):
| (7.8) |
Here, , , , and the equality (6.37) was used.
Since for arbitrary , we have , (7.3) gives that
If we denote by and the first and the second integrals in the right-hand side of the above equality, correspondingly, we can apply (2.64)(c) and (2.64)(d) to get and
| (7.9) |
Let . Then implies and using (7.2) we get
from which
To obtain the final estimates of , we need to deal with the variable lying in but not in . Hence, we transform (7.10), taking into account that due to -periodicity of we have , where for all , in view of (5.95)(a). Namely,
from which
By applying for the three expressions in (7.8) the obvious consequence , , of (5.13), , the left-hand side inequality of (7.1), (2.25)(a),(c) and (7.12), we obtain in Section 10.5.1 the following properties.
Lemma 7.1
. Let , , and the function be defined as in Theorem 6.4. Then
| (7.16) |
, for arbitrary and .
By (6.47), for every . At the same time, for any and we have and , in view of (5.95)(b) and (6.50), correspondingly. Applying (6.47) once more, we get . Since of the variable is continuous on we derive from (7.16) and (7.4) the next assertion.
Corrollary 7.2
. Let , , and the function be defined as in Theorem 6.4. Then
| (7.19) |
7.2 . Main inequalities.
It follows from (6.9) that for arbitrary and we have
By Theorem 6.4, for every the function can be analytically extended to and in a second step, by (6.13), to -periodic holomorphic function on , which equals on . As a consequence, by Lemma 3.1, we obtain from the above formulas that
from which it follows that
| (7.24) |
for and . Since we can apply (7.19) to estimate the integral in the right-hand side of (7.24) as follows
The corresponding estimates of can be derived from the explicit integral formula written after (1.21):
| (7.26) |
The first immediate consequence of the obtained estimates and (6.9) is the possibility of expressing the functions , , for any in the form
| (7.27) |
because, by (7.25) and (7.26), both series on the left-hand sides of the equalities in (6.9) turn out to be holomorphic on . It follows from (6.8), the identity , (see (2.41)), and the asymptotics as , that for all and , the function
| (7.28) |
of the variable is the unique primitive of which has limit as . Then we derive from (6.9) that
and since both series here for each are holomorphic on , in view of (7.25) and (7.26), we conclude that for any and the function has analytic extension from to and the resulting extension satisfies
| (7.29) |
In comparison with (7.28), the formula (7.31) determines the values of for arbitrary and , and, it can be easily calculated that
This relationship enables us to apply the reasons as in the proof of Lemma 6.1 to obtain for arbitrary and that
| (7.32) |
8 . Interpolation formula for the Klein-Gordon equation
8.1 . Hyperbolic Fourier series in .
In view of (6.13), (6.28) and (6.16), for arbitrary fixed and we have for any , where . Then for each it follows from (6.12), (6.16), (6.27) and (6.13) that
and since the functions appearing on the two sides of this identity are holomorphic in we obtain, by the uniqueness theorem for analytic functions (see (con, , p. 78, Theorem 3.7(c))), that
| (8.1) |
By substituting the identities (7.27) and (7.29) into (8.1)(with ) and in (8.2), respectively, we see that for arbitrary and the following identities hold:
| (8.3) |
and
| (8.4) |
By multiplying the latter equality by a function we integrate it with respect to and apply well-known property of functions in (see (ko, , p. 116)), together with the estimate (2.48) and the identity (see (2.52))
to obtain that enjoys the representation (see (1.5))
| (8.5) |
We now extend this representation to a larger class of functions. According to (gar, , p. 60, Corollary 3.4), if and only if . Since then for any we can expand as in (8.5). We then apply the estimates (7.25), (7.26) together with the Lebesgue dominated convergence theorem (nat, , p. 161) as to get the following representation for functions in a weighted Hardy class (cf. Theorem 3.1 in (bon, , p. 14)).
Theorem 8.1
. Let . Then
where
and , , , .
In view of Jordan’s lemma, for any the function is bounded and holomorphic in , and we can calculate that
| (8.6) |
By multiplying (8.4) by this function and after integrating over , we obtain as above,
for all and , where
| (8.7) |
Here, the change of variable and the symmetry property (1.19) entail that, for arbitrary ,
| (8.8) |
while, as a consequence of (3.20), we have
| (8.9) |
8.2 . Conjugate hyperbolic Fourier series.
For arbitrary , the series in the right-hand side of the equality (1.9) is called conjugate hyperbolic Fourier series of , where the coefficients are defined as in (1.7). Taking account the estimates (7.25), the result (1.9) can be improved as follows.
Theorem 8.2
We now apply Theorem 8.2 to obtain the conjugate hyperbolic Fourier series expansion of the Poisson kernel
| (8.11) |
This allows us to expand the harmonic extension to the upper half-plane of a given given by convolution with the Poisson kernel in the form
This formula can be understood as the regularization of the hyperbolic Fourier series (1.5) of found by considering the harmonic extensions of the basis functions.
8.3 . Density in Hardy classes.
By manipulations similar to those employed in the proof of Corollary 3.3 in (gar, , p. 59) we obtain in Section 10.6.1 the following property of the Hardy class .
Lemma 8.3
. The linear subspace is dense in .
In view of (1.9), every , can be expanded in an absolutely convergent conjugate hyperbolic Fourier series, all whose coefficients with non-positive indexes are zero, as follows from (1.6) and the known properties of the functions from the class (see (gar, , p. 88, 2(iv))). It now follows from (4.1) and (1.18) that the following property holds.
Theorem 8.4
. The function system is complete in , while the functions form a complete system in .
8.4 . Interpolating functions and interpolation formula.
We apply Jordan’s lemma and the residue theorem in the same way as for the proof of (8.6), and obtain
| (8.12) |
for all , . We proceed and apply this identity together with (8.6) to the formulas (2.52) written in the form
| (8.13) |
and obtain from (8.7), by using (2.48), (3.36)(a) and (3.29)(a), that the integral representations
| (8.14) |
hold for all , because for all , as we see from the estimates (7.25) and (7.26). In addition to the symmetry property (8.9), we observe that by substituting (8.13) into (8.7), while taking into account (8.12) (interchange in orders of integration is justified by (3.36)(a) and (3.29)(a)), we arrive at
| (8.15) | ||||
and hence, since is continuous on for each , we find that
| (8.16) |
This property also follows directly from (see (4.1)) and the equality (8.15). In Section 10.6.1 we prove the following assertion.
Theorem 8.5
. Let be given by (8.7). Then , and, in addition, for each , the restriction of the function to the quadrant extends to all of as an entire function of two variables. At the same time, we have
| (8.19) |
The function satisfies
| (8.20) | |||||
| (8.21) |
while for any we have
| (8.22) | ||||
| (8.23) |
where , , , are the modified Hankel function satisfying as for each .
For , let be defined as in (1.1), i.e.,
| (8.24) |
Then solves the Klein-Gordon equation in the sense of distribution theory on . We recall that is a solution of the Klein-Gordon equation on a given open subset in the sense of distribution theory if and the equality holds in the sense of an equality for the linear functionals on test functions , the compactly supported -smooth functions mapping to (cf. (hor, , pp. 14, 34)). Given that our primary interest is in solutions of the form , which are continuous on , it is more convenient to use alternative definition which is equivalent to the definition above for continuous solutions of the Klein-Gordon equation.
Definition 8.6
. Let be an open convex subset of . We say that is a continuous solution of the Klein-Gordon equation on if and
| (8.25) |
for all , where and .
We readily verify that (8.25) holds for and , and, consequently, is a continuous solution of the Klein-Gordon equation on .
Next, let us assume that meets the condition (8.10) of Theorem 8.2. In view of the identities (1.10), this is the same as requiring that (1.11) holds. Then the conjugate hyperbolic Fourier series (1.9) of can be substituted for into the integral in (8.24) to get the equality (1.12) by taking into account the identities (1.15). As a consequence, we obtain the following interpolation formula for the solutions of the Klein-Gordon equation of the type (8.24).
9 . Extension of Theorem A
In this section, we derive the following extension of Theorem A. We make an effort to present a direct proof which does not rely on ergodic theory, and which also covers the instance of Theorem A (when in (9.3) below). This makes the approach direct, but we should mention that there are shortcuts available if we were to abandon this aim.
Theorem 9.1
. Let , , , and , . Assume that
| (9.3) |
where . Then .
The analogue of this assertion for the biorthogonal system , , , , can be formulated as the following theorem.
Theorem 9.2
. Let . Suppose that
Then .
multiply it by the function and integrate over , to obtain from the given assumptions that
It follows that the function , which is holomorphic in , must be constant. Since automatically implies that as , we find that for all . By the known characterizations of the space (see (gar, , p. 88, 2(ii))), we obtain that . Theorem 9.2 follows.
9.1 . Proof of Theorem 9.1.
We observe that an application of the finite difference over of order and to (9.3)(a) and to (9.3)(b), respectively, gives (see (mil, , p. 29))
or, in the notation , , where by a change-of-variables, the conditions may be written as follows:
From this we obtain that
| (9.6) |
for almost all because the system is complete in . Since the right-hand side series in (9.6) represent integrable functions on we conclude that the left-hand side functions in (9.6), being obviously integrable on , are also integrable on , and hence
If for arbitrary we introduce
| (9.7) |
then
from which
and thus
Then it follows from (9.6) that
i.e.,
| (9.8) |
where it can be easily seen that for arbitrary in the space there exists the limit
| (9.9) |
In view of , , , the operator (9.9) for arbitrary Borel set possesses the following essential property
or, in the notation (5.1),
| (9.10) |
because the sets , , are disjoint. Since we can successively apply (9.10) to get
| (9.13) |
where for the Lebesgue measure on the real line and a Borel set after the similar manipulations we obtain
| (9.14) |
because, in view of , ,
But according to (10.77), , and hence, we obtain from
and (9.15) that
and, consequently, (9.14) implies that
| (9.16) |
Iterating (9.8) gives
which together with (9.13) lead us to the conclusion that
| (9.17) |
But in view of (9.18) below and (9.16) we have , which by and (9.17) yields that for almost all and for any . Definition (9.7) of the functions gives that and for almost all . Thus, for almost all and Theorem 9.1 follows.
We supply the following auxiliary lemma which was referred to in the proof of Theorem 9.1.
Lemma 9.3
. Let be defined as in (1.22). Then
| (9.18) |
Proof of Lemma 9.3. We first obtain that
| (9.19) |
Assume that there exists such that
Since is nondecreasing on (see (hed1, , p. 1715, Proposition 3.7.2)) then
For arbitrary and we obviously have
| (9.20) |
and, in view of the inequality
and of nondecreasing property of on , we get
As a consequence, we derive from (9.20) that
Iterating this procedure for , we obtain
i.e.,
Hence,
| (9.21) |
In view of (9.13), for any the set contains at least all irrational point of . Therefore its Lebesgue measure is equal to . Then, by virtue of (9.14),
as . This contradiction proves (9.19).
Next, to establish (9.18), we appeal to the formula (9.15) with . In view of (10.66), for arbitrary we have
| (9.24) |
where . These relationships mean that we can apply conclusion (10.71) to the finite collection of numbers with , according to which,
In particular,
for all . Hence,
| (9.25) |
10 . Proofs
10.1 . Proofs for Section 2
10.1.1 . Proofs of Lemma 2.1 and (2.31)(b).
The statement of Lemma 2.1 is immediate from the relationship
proved in (bh2, , p. 608, (3.15)) and from the identities
where and .
from which
because
Inequalities (2.31)(b) is now immediate from and from
10.1.2 . Proofs of (2.50) and (2.51).
For the integrand in the formula for we have
we obtain
Hence,
which proves (2.50).
and therefore
which completes the proof of (2.51).
10.1.3 . Proofs of Theorem 2.2 and Corollary 2.3.
and therefore
which proves the second equality in (2.55). Similarly,
for arbitrary , and hence, by (2.25)(g), we obtain
which completes the proof of the first equality in (2.55).
Prove now (2.56) and (2.57). For it follows from and (2.39) that and because . These two inequalities together with (2.55) yield the validity of (2.56) and (2.57) for . If in (2.56) and in (2.57), we similarly get their validity by using inequalities and which follow from (2.39), in view of . The proof of (2.56) and (2.57) is completed.
where, by virtue of (2.32), the both functions of in the right-hand sides of (10.7) strictly decrease on the interval from to . This proves (2.59) and (2.60).
The property (2.58) is immediate from (2.40). Furthermore, (2.64)(a) is a simple consequence of (2.58) and (2.59).
provided that . But if then by (2.59) and we get and therefore
which together with (10.8) completes the proof of (2.64)(c). Next, by (2.41), (2.60) and (ber1, , p. 325, (i)), for and we obtain
10.1.4 . Proofs of Theorem 2.6 and Corollary 2.7.
For and with , we have
Besides that,
which is strictly less than if , but if then and
provided that and the equality here is strict if or , or . Thus, the following properties hold.
| (10.13) | |||||
| (10.22) |
According to the definition (5.1), for and we have
| (10.23) |
where
| (10.24) |
where (10.30)(b) with follows from (10.30)(a) with , applied to which by (10.24) and (10.22) satisfies and . Introduce the notation
| (10.33) |
Furthermore, by Lemma 2.5,
| (10.40) |
It has been established in Lemma 5.7 that for each the set is countable and cannot have the limit points in . That’s why to prove the statement of Theorem 2.6 it suffices to show that the imaginary part of each number from is strictly less than , i.e.,
| (10.41) |
which obviously coincides with the property (2.84).
Assume that the number in Theorem 2.6 belongs to . In the notation , it follows from (10.36) that and therefore (5.63) yields
| (10.42) |
Let in Theorem 2.6 belong to and . By virtue of (10.36) and (10.39), and we deduce from (5.63) that
| (10.43) |
Let finally in Theorem 2.6 belong to and . Then ,
| (10.44) |
and (10.36), (10.39) and (10.40) imply that and
| (10.45) | |||||
(see (5.27)). In accordance with (5.63), we get
| (10.46) |
Here , for every , where if and only if . Applying (10.30)(c) and (10.30)(d) to with , we obtain
| (10.48) |
10.2 . Proofs for Section 3
10.2.1 . Proofs of Lemmas 3.2 and 3.4.
where , and therefore, there exist the real numbers , , such that
where . Then there exist the real numbers , , satisfying
and , , such that
Hence,
where obviously, and . Lemma 3.2 is proved.
where , , , are the real numbers such that for each . Hence, in the notation , , of (3.14), it follows that
To prove (3.16), observe that as (see (bh2, , p. 609, (4.6))), which yields, in view of Lemma 2.5, that as . So that for every one can find such that for all and (3.9) can be written as follows
| (10.49) |
By choosing a positive number less than we deduce from (10.49) that
Therefore for every the series in (3.6), written in the form,
converges uniformly over all and we can take the limit as to get, in view of (2.41) and arbitrariness of , the following equivalent form of (3.16)
| (10.50) |
It follows from the functional equation
(see (bh1, , p. 48, (A.14n))) that
| (10.51) |
By using (3.2),
we obtain
| (10.52) |
10.3 . Proofs for Section 4
10.3.1 . Proofs of Lemmas 4.1 and 4.2.
We first prove Lemma 4.1. The properties Lemma 4.1(1),(2),(3) are immediate from (3.37), (3.29), (3.33) (4.1) and (1.19). To prove Lemma 4.1(4) for positive integer we change the contour of integration in (3.37) to the contour which passes from to along , from to along and from to along . By using the periodicity of and , we obtain
and therefore, for any positive integer it follows that
Letting and using the identity (see (gra, , p. 44, 1.422.4))
we derive from the first of the latter two equalities and (3.21) that
and from the second one and (3.18) that
This completes the proof of Lemma 4.1(4) for positive integer . But according to (1.18), for any we have
and, by (1.19),
from which for all , in view of the continuity of . Then
Next, we prove Lemma 4.2. The properties Lemma 4.2(1),(2),(4) are simple consequences of (3.20) and (2.52). To prove Lemma 4.2(3) we change the contour of integration in (2.52) similar to that of in Section 10.3.1 and by the periodicity of we get
and therefore, for any positive integer this yields
we derive from the latter equality that
10.4 . Proofs for Section 5
10.4.1 . Proofs of (5.9) and (5.32).
We first prove (5.9). In the notation
| (10.53) |
where and such that , we obviously have
| (10.56) |
for some , and every .
observe that for any , and therefore it follows from that
| (10.59) |
Successive applications of (10.59) to (10.57) prove (10.58) and the right-hand side inclusion of (5.9) as well.
For and from (10.57) introduce the matrices
| (10.62) |
whose elements obviously satisfy
| (10.63) |
and
| (10.66) |
and ()
| (10.68) |
We prove that
| (10.69) |
For arbitrary positive integer , real numbers satisfying and nonzero integers , we define the following collection of real numbers
| (10.70) |
We state that
| (10.71) |
We prove , , by induction on . For such an inequality holds because , and hence, . This also completely proves (10.71) for . If we assume that for all for some . Then , i.e. . By induction, we conclude that (10.71) is true.
It remains to examine in (10.69) the case . By setting first , , , , , , , in (10.70) and then , , , , , , , we deduce from (10.66) and (10.71) that
| (10.72) |
where the last inequality follows from and .
For positive integer let . Introducing by the formulas (10.62) matrices corresponding to , we get
Applying the transpose operation to , we obtain
from which,
| (10.76) |
and therefore, by arbitrariness of , we deduce from (10.73) that
| (10.77) |
Together with (10.72) and (10.68) this proves the required inequalities , in the left-hand side of (5.9) (cp. (boc, , p. 4, Lemma 2)).
Besides that, the relationships (10.66) written for and give
where the inequalities (10.73) have been used. Since is arbitrary integer satisfying , we get
| (10.78) |
Indeed, (10.72) and (10.66) yield that for each and hence, (10.81)(a) holds. To prove (10.81)(b) assume to the contrary that there exists such that . Then which contradicts (10.78). This contradiction proves (10.81)(b). Observe, that (10.81) for yields
| (10.82) |
It follows from (10.77) that and therefore we obtain the inequalities of (5.32)(a) , because (10.63) gives
| (10.86) |
At the same time, for we have , as follows from (10.85) and (10.66). This proves (5.32)(b) for . But if then we deduce from (10.63) that the sign of the number is equal to , in view of (10.77) and (10.81)(b). This proves (5.32)(b) for . Since (5.32)(c) is immediate from (10.69)(a) and (10.85), the proof of (5.32) is completed. Hence, for and , the hyperbolic quadrilateral has the shape shown in Figure 3, provided that (in view of (10.81)(b), yields ).
10.4.2 . Proofs of Lemmas 5.3 and 5.4.
We first prove Lemma 5.3.
(a) Assume the contrary, i.e., there exist , , , , , , such that , which by (10.57) can be written as follows (see (5.27))
| (10.87) |
Without loss of generality one can consider in (10.87) that , because otherwise, we can apply to the both parts of (10.87) to get the similar relationship for and of lower dimension and to get , if , or
| (10.88) |
where the left-hand side inclusion follows from (5.9). But (10.88) cannot hold since . So that we assume everywhere below that .
Let in (10.87). Then , where . Since then can take only two values or . If then there exists such that and , which yields and , by virtue of (5.27) and (5.28). But then which contradicts . Therefore , and it follows from (10.87) that
if . But none of these equations can hold because their left-hand sides belong to , in view of (5.9), while their right-hand sides belong to . Lemma 5.3(a) is proved.
(d) Let be such that if (see (5.27)). As noted before Lemma 5.3, is the roof of . But is also a part of the boundary of
and since
for , we get that
According to what was stated before Lemma 5.3, this means that is the lower arc of , if . At the same time, for is the lower arc of . Lemma 5.3(d) is proved. Lemma 5.3(b) and (c) follow by similar arguments.
We prove now
| (10.91) |
by induction on , where and are defined as in (10.66). In view of (10.66), , and therefore (10.91) holds for . In addition, (10.91) is proved for . Assume that , and (10.66) is true for . In view of (10.72), we have
Then and by (10.66) we get
which proves the validity of (10.91) for . By induction, we conclude that (10.91) is true for all . Combining (10.91) for , (10.90) and (10.89) gives Lemma 5.3(e) and completes the proof of Lemma 5.3.
Next we prove Lemma 5.4. Since for every it suffices to prove that for any point in there exists such that contains this point.
Let , and assume that for each . We denote by the number . Then by (5.20), , because . Therefore for arbitrary there exists a minimal positive integer such that
| (10.92) |
where, in accordance with (5.14),
| (10.93) |
Since for each we deduce that
| (10.94) |
and if for some , then (10.93) yields
In view of (5.12), this means that and therefore . Then , by virtue of (10.93). Hence, in addition to (10.94), we have
| (10.95) |
Thus,
| (10.96) |
which completes the proof of Lemma 5.4.
10.4.3 . Proofs of Theorems 5.5 and 5.9.
and since we deduce from (5.7) that the union of all boundary points of the sets composing the Schwarz partition (5.40) of , satisfies
Inverse inclusion is immediate from the invariance of under any transform and obvious properties , and . This completes the proof of Theorem 5.5.
Next, we prove Theorem 5.9. It follows from (5.81) and (5.83) that the union of all boundary points of the sets composing the even rational partition (5.86) of is equal to the union of the roofs of over all and their shifts on any even integer. But by and (5.7) this yields that coincides with the orbit of with respect to . Theorem 5.9 is proved.
10.4.4 . Proof of Lemma 5.7.
According to the partition (5.38), we have
| (10.98) | ||||
Let be arbitrarily prescribed and fixed number. It follows from Lemma 2.4 that
| (10.102) |
Fix now an arbitrary , which, in the notation (5.27) and (5.29), is associated with and such that and , where is the Möbius transformation defined as in (5.1). Since is injective on , we get
| (10.103) | ||||
where, in view of (5.58),
If , then and, in view of Lemma 2.4, the solutions of the both equations in the right-hand side of the latter equality are unique. Hence,
| (10.110) |
If then , while , by Lemma 2.4 and Theorem B. So that there is no solutions of the equation in the right-hand side of (10.104) for odd . But for even such an equation has the unique solution , if , and , if . Or, what is the same . Thus,
| (10.113) |
10.5 . Proofs for Section 7
10.5.1 . Proof of Lemma 7.1.
10.6 . Proofs for Section 8
10.6.1 . Proofs of Lemma 8.3 and Theorem 8.5.
To prove Lemma 8.3 for arbitrary we introduce the function
where (see (bak, , p. 658, (1.17), (1.18))) ,
For an arbitrary let
| (10.117) |
Then for any the relationships
yield that
| (10.118) |
Since by (gar, , p. 88, 2(iv)) we have for every then the identity
implies
| (10.119) |
by virtue of (10.118) and (gar, , p. 88, 2(iv)). For arbitrary we introduce the Fourier transform of as follows
| (10.120) |
where denotes the class of all nontangential limits on of the uniformly bounded and holomorphic on functions. Integration by parts gives
It follows from the inequality , (see (abr, , p. 70, 4.2.38)) that for arbitrary the following inequality holds
| (10.122) |
We introduce the notation
Now, for arbitrary and it follows from (10.120) that
where
and, in view of (10.122),
while
So that
where the right-hand side tends to zero as because and the translation is continuous in (see (gar, , p. 16)). This together with (10.121) completes the proof of Lemma 8.3.
To prove Theorem 8.5 we notice that in accordance with (8.14), for each the functions and on the quadrant coincide, correspondingly, with the functions
which both are entire functions of two variables because (2.48) and (2.47) can be written as
| (10.123) |
for all . This proves the first assertion of Theorem 8.5.
To prove (8.21) and (8.23) observe that the following estimates have been derived after (4.5) for ,
| (10.124) |
where and the value of has been used from (7.2). Then, by using the estimates (2.47), (2.48), for the parametrization , , we derive from (8.14) and (10.124) that
So that
| (10.127) |
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Remark to the References. The results of this preprint are based on several assertions which may be found in abr , and , bh2 , bh1 , (bak, , p. 665), ber1 , con , erd1 , gar , gra , (hed1, , p. 1715), ko , lav , nat , olv , rud , vlad and whi . The other cited books and articles are listed for completeness.
For example, the monograph law is cited on the page 2.44 in connection with the Landen transformation equations (2.44). However, these equations have self-contained proofs on the pages 5–6. Similarly, in connection with the reference to (lop, , p. 303) on the page 5.8 we supply a self-contained proof of (lop, , p. 303, Proposition 2) on the pages 27–28. Finally, the references to Lemma 2 of (cha, , p. 112) on the pages 5.3 and 5.3 are complemented by a self-contained proof of this lemma on the pages 19–21.
Moreover, in (hed1, , p. 1715) we use only Proposition 3.7.2 which establishes the elementary property of the operator that it preserves the properties of functions on to be even and convex. Also, we need (bak, , p. 665) to write explicitly the value of the integral of the simplest test function. At last, when using a property of the theta functions needed in the preprint, we refer to bh2 and bh1 because all the basic properties find simple proofs there with the help of elementary integrals, Liouville’s theorem, Morera’s theorem, Riemann’s theorem about removable singularities, and elementary properties of the Schwarz triangle function . All the other facts used in the preprint without proofs are from the fundamentals of real and complex analysis found in, e.g., abr , and , ber1 , con , erd1 , gar , gra , ko , lav , nat , olv , rud , vlad and whi .
A . Supplementary notes
A.1 . Notes for Section 1
1
It follows from the two equalities preceding (1.6) that for arbitrary and we have
Thus, the conditions in (1.8) imply that the sequence
converges and
satisfies
This means that defines a continuous linear functional on and therefore (see (vlad, , p. 77)). In conclusion, the series
converges to in the space .
2
and (2.20), that
where , and then, by using
| (A.14a) |
we can apply to the latter equality the function to get
| (A.2a) |
and therefore
| (A.2b) |
and, substituting these expressions in (A.2b), we get
from which
| (A.2c) |
Since , we can set in (A.2c) and consider the equation
and therefore
Since
and
we can replace in (A.2c) by
and to get
where
and hence,
| (A.2d) |
which proves (1.21).
A.2 . Notes for Section 2
4
Since (see (bh2, , p. 598, (1.26)))
| (A.4d) |
for the principal branches of the quadratic and of the fourth roots it makes possible to consider three functions
which are holomorphic in and in accordance with (A.4c),
| (A.4e) |
Therefore
Here each of the set is relatively closed in , i.e., is closed for every compact subset of , because , , . In view of the Baire Category Theorem (roy, , p. 159), for each there exists a number such that , , if , and the set contains some neighborhood of at least one interior point of . Applying the uniqueness theorem for analytic functions (see (con, , p. 78)) we obtain for all and therefore
| (A.4f) |
5
We prove (2.44). In view of the uniqueness theorem for analytic functions (see (con, , p. 78)), it is enough to prove that
| (A.5c) |
because , , as follows from (A.4d).
We apply the approach suggested in (bh1, , p. 24). According to this approach, it is necessary to introduce the following three functions
| (A.5d) |
which are holomorphic in and . And then to study the values of on the two sides of the cuts along and along , and their behavior near the points .
We first look on their values on the two sides of the cut along . Since by (bh2, , p. 19, (6.8)) we have
which yield (see (bh2, , p. 56, (A.18a)))
then
and we derive from (2.66),
that for every we have
| (A.5e) | ||||
| (A.5f) |
Applying to these relationships the Morera theorem (see (lav, , p. 96)) we obtain that
| (A.5g) |
and therefore could be a point of an isolated singularity for all functions (see (con, , p. 103)). We prove that actually is a point of a removable singularity for these functions. To prove this, we apply the Riemann theorem about removable singularities (see (con, , p. 103)), according to which is a point of a removable singularity of , where , if and only if . For any the latter equality obviously follows from more strong property, , and this in turn is equivalent to , because and therefore is continuous on . According to (bh2, , p. 609, (4.2), (4.5), (4.6)),
and since
as , we obtain the existence of the following limits
Thus,
| (A.5h) |
We now look on the values of on the two sides of the cut along . For arbitrary and , by using the identities,
(see (bh1, , p. 57, (A.18b)(b), (A.18b)(c); p.58, (A.18e)(c))) we obtain, for every ,
and derive from (2.67),
that
| (A.5i) | ||||
| (A.5j) |
Applying the Morera theorem (see (lav, , p. 96)) to these relationships we obtain
| (A.5k) |
where the function
| (A.5n) |
is holomorphic in as follows from (A.5h), theorem about analyticity of the composition of two holomorphic functions (see (con, , p. 34)), and the fact that maps conformally onto . The latter property is the consequence of the fact that the function maps conformally onto (see (con, , p. 46)). Therefore the inverse mapping maps one-to-one onto . Then , , and (A.5n), for arbitrary imply
which by (A.5i) means that
and hence the Morera theorem can be applied to to get . The right-hand side inclusion of (A.5k) is completely proved.
We prove now that and are the points of a removable singularity for and , respectively. According to (bh2, , p. 609, (4.2), (4.4)),
| (A.5o) |
because in view of , it follows from that and , which means that . Then the asymptotic equalities (see (2.25)(a),(b),(c)),
as , (and hence, ) yield
Hence, by (A.5o), we have, for ,
| (A.5p) |
In view of the continuity of on , as follows from (A.5k), the limit of as exists if and only if there exists the limit of as , and we derive from (A.5p), and , the existence of the following limit
Together with (A.5p) for and (A.5k) this relationship yields that and are the points of a removable singularity for and , respectively, and so and are entire functions, i.e.,
| (A.5q) |
We prove now that the modulus of and are uniformly bounded on by establishing the existence of the finite limits of and as .
Let approaching from one of the half-planes . Then (see (bh2, , p. 609, (4.2), (4.3))) and, in view of (2.16), . The latter equality yields (see (bh1, , p. 24, item 2)) and by manipulations similar to those employed in the proof of (A.5o), for arbitrary we get
| (A.5r) |
By using the relationships (see (bh1, , p. 58, (A.18e), (A.18d)),
we obtain, for arbitrary , the existence of the following limits
Hence, for arbitrary there exist the limits
| (A.5s) |
Since by (A.5q) we have then it follows from (A.5s) with that as and we obtain that the entire function is bounded on while , also according to (A.5q). The Liouville theorem (con, , p. 77) yields for every and, in particular, for every . Then and (A.5d) imply the validity of (A.5c) for and completes the proof of (2.44)(c).
Since if and only if and, in accordance with (A.5q), we have , then the limit of as exists if and only if there exists the limit of as . But in the latter case and . In view of the definition (A.5n) of and the properties (A.5s), we conclude that as and therefore the entire function is bounded on while , according to (A.5q). The Liouville theorem (con, , p. 77) gives for every , which by (A.5n) yield that for every and for every . This implies that for every because maps conformally onto . Then and (A.5d) yield the validity of (A.5c) for and which completes the proof of (2.44)(a) and (2.44)(b). The Landen transformation equations (2.44) have been completely proved.
6
Let . Since it is possible to use the principal branch of the square root for and as well. Then by (2.21) and (2.26) we get
| (A.6a) |
Then, by virtue of (2.44)(b), we get
from which
| (A.6b) |
Substituting here (2.38)(b) written in the form
| (A.6c) |
we obtain
| (A.6d) |
and after squaring this identity the constraint can be dropped,
| (A.6e) |
and we deduce from (A.6d),
| (A.6f) |
Or, what is the same,
| (A.6g) |
which coincides with the quadratic transformation (3.1.10) with of (and, , p. 128) for the hypergeometric function .
7
The Landen relationships (2.44)(a)–(c),
| (A.7d) |
can obviously written as follows
| (A.7h) |
from which, by (2.26), we get
i.e.,
| (A.7i) |
Therefore
we obtain
from which, taking account of (2.21), for arbitrary we get
i.e., in the notation (2.46),
| (A.7j) |
8
The right-hand side equality (2.45) written for
where for all , makes it possible to write
i.e., by (2.21),
| (A.8a) |
that proves equality in (2.47).
By using (2.31)(b) written in the forms,
we derive from (A.8a) that
from which we obtain the inequality in (2.47),
| (A.8b) |
9
The Landen relationships (2.44)(c) together with (2.25)(g) and (A.18e) of (bh1, , p. 58) for any give
which by (2.25)(a), (c),
lead to
and therefore
| (A.9c) |
Making here the change of variables , we obtain
| (A.9f) |
When , ,
and (A.9f) yields
Hence,
Here
while, by (ber1, , p. 325, (xii)),
and therefore
But
and so
where
which completes the proof of (2.48).
A.3 . Notes for Section 3
10
We prove the statements of Lemma 3.1.
Suppose for the moment the assertion of Lemma 3.1 holds with . Then we would apply that statement to the new function and derive that Lemma 3.1 holds in fact for any . Hence it suffices to obtain (3.1) in the case only. Then our assumptions are that the functions and are both holomorphic in and -periodic.
For , , , we have
| (A.10a) |
and since we obtain that for every , where
Here, if we write for each , we find
and consequently, extends continuously across . From Morera’s theorem (see (lav, , p. 96)) we conclude that . For this completes the proof of the second assertion of Lemma 3.1. As for , we obtain that admits the Laurent expansion (see (con, , p. 107)) , , which is absolutely convergent in . The change of variables , , , , gives, by using (A.10a), that , and, moreover, that
Both and the Fourier series on the right-hand side are continuous on and periodic with period . Hence they must be equal for all and, consequently,
14
Lemma A.1
. Let , and be the Möbius transformation (con, , p. 47, Definition 3.5) such that
| (A.14a) |
Then
| (A.14b) |
Proof. The boundary of the image of under the Möbius transformation
according to the known property of the Möbius transformations to map circles onto circles (con, , p. 49, Theorem 3.14) and to the known open mapping theorem (con, , p. 99, Theorem 7.5) of nonconstant analytic functions to map open set to open set, consists of four circles and therefore four vertices of are transformed into vertices, i.e.,
Since then if then
| (A.14c) |
also map onto itself. Therefore it suffices to consider two cases
In the first case we get and therefore one may assume that . Then , i.e. it should be
we get three cases , , , where correspondingly
and since is not acceptable because , we obtain .
For the second case we have , i.e.
and it follows from that
and therefore it should be
We get three cases
Thus, and taking account of (A.14c) we complete the proof.
A.4 . Notes for Section 4
16
In accordance with (4.5) for we have
| (A.16a) |
For the parametrization , , taking into account
we obtain
| (A.16c) |
Then we deduce from (3.37) and
that for
| (A.16d) | ||||
| (A.16e) |
where
Thus,
from which
| (A.16f) |
where
where
So that
and hence,
| (A.16g) |
17
Using (gra, , p. 44, 1.422.4), we obtain the validity of the left-hand side equality in
while the validity of the right-hand side equality here follows from the following identities
A.5 . Notes for Section 5
19
Let . Then by Lemma 2 of (cha, , p. 112) the transform can be represented as a superposition of a finite number of the degrees of the transformations and (uniqueness of such representation has not been proved by Chandrasekharan in cha ). In the notation of (10.53) and in view of (10.56), this means that either or there exist , and such that (see (bh1, , p. 63))
and by setting in the above equalities to each of four possible values we obtain that
which completes the proof of (5.3).
20
Lemma 2 of (cha, , p. 112) can be sharpened as follows.
Theorem A.1
. For arbitrary there exists a unique collection of numbers , , and such that
| (A.20a) |
where
, , and denotes the set of all matrices with integer coefficients satisfying .
The fact that is invariant under multiplication of matrices follows from the possibility of a termwise multiplying the congruences (see (apo, , p. 107, Theorem 5.2(b))) and , while belonging to of each its inverse matrix is a consequence of the general formula
| (A.20b) |
As well as , the set is also invariant with respect to the transpose operation
| (A.20c) |
Everywhere below we use the notation .
Before proving Theorem A.1, we first establish several special properties of matrices from the set . Each column and each row of any matrix from the set contains two numbers of different parity and therefore they cannot be equal to each other. In fact, a much stronger property of these matrices takes place.
Lemma A.2
. Let . Then the value of is attained on only one element.
Proof
. Suppose to the contrary that is attained on more than one element. Since the pairs , , and consist of numbers with different parity then the only one of two following cases can take place
| (A.20f) |
If (A.20f)(1) holds then
| (A.20g) |
from which and therefore , i.e.,
| (A.20h) |
We say that is modulo isotonic if
| (A.20i) |
Lemma A.2 can be essentially sharpened as follows.
Lemma A.3
. Let . Then among four matrices
| (A.20j) |
each of which belongs to , there exists exactly one modulo isotonic matrix.
Proof
. Four matrices form the group with the matrix multiplication as a group operation, and therefore one of the matrices from (A.20j) can belong to if and only if contains the matrix . It readily follows from Lemma A.2 and (A.20j) that there exists a unique ordered pair of numbers such that the matrix at the intersection of its first row and first column contains exactly that element of which has the smallest absolute value. By designating the elements of the transformed matrix by the same letters, we have
| (A.20k) |
To prove that is modulo isotonic, it is sufficient to show that and .
Assume first that
| (A.20l) |
By virtue of the different parity of and , we have and therefore , i.e., and , which yields . Then implies and therefore . Our assumption gives . Finally, we obtain and . This holds if and only if for some . By solving the equation we get which contradicts . This contradiction proves .
If we assume that then the conjugate matrix also belongs to and satisfies the conditions (A.20k) and (A.20l). By application of the above reasoning we conclude that for some . Then and similarly to the above , which contradicts . This contradiction proves and that is modulo isotonic. But if one of the matrices in (A.20j) is modulo isotonic then no other one in (A.20j) can have this property because all other matrices contain the unique element of with the smallest absolute value at the wrong place. Lemma A.3 is proved.
The next assertion is immediate from Lemma A.3.
Corollary 1
. Let satisfy and . Then and , i.e., is modulo isotonic.
The following result shows that any modulo isotonic matrix from the set is uniquely determined by its second column.
Lemma A.4
. Let two matrices and belong to and be modulo isotonic. Suppose that and . Then .
Proof
. Assume that two different modulo isotonic matrices in satisfy
| (A.20m) |
In view of (A.20i), and it follows from , that
| (A.20n) |
But (A.20n)(1) and (A.20n)(2) implies that divides and since we get that divides . Hence, taking account of (A.20o) and (A.20n)(3), there exists such that . Substituting this in (A.20n)(1) we obtain , where we can divide by nonzero number and derive from (A.20i) that
| (A.20p) |
It follows from (A.20p)(1) that and since by (A.20n)(3) is nonzero we conclude that the condition (A.20p)(3) can be replaced by
| (A.20q) |
According to the definition , we have only two possibilities
| (A.20t) |
Lemma A.5
. Let be modulo isotonic (see (A.20i)). Then there exists a unique collection of numbers , and such that
| (A.20u) |
Proof
. By (A.20i) and the definition of , and are nonzero integers of different parity, and it follows from that . By applying Lemma A.6 to the rational number we obtain the existence of the unique and such that
where
21
We prove Lemma 5.3(b). The lower arc of is a part of the boundary of
and also
As stated before Lemma 5.3, this means that is the roof of .
Let . Then the lower arc of is a part of the boundary of
and
22
We prove (5.19). (a) We have if and only if there exists such that
By using (5.12), according to which
we obtain by (5.14),
Since is arbitrary, (5.19)(a) is proved.
(c) In view of (5.6),
23
We prove (5.26). It follows from
and from
| (5.9) |
that (5.23) holds because
Therefore
and
what was to be proved in (5.26).
25
We prove directly that the right-hand side set in (A.25a) equals to the right-hand side set in (A.25b). Everywhere below we use the notation
and the following identities
| (A.25j) |
which hold because
and
26
We prove (5.51). The set in the right-hand side of (5.51) equals to , in view of the definition (5.8). The relationship (5.3) yields that and since
| (A.26c) |
we deduce from (5.9) that
| (A.26g) |
27
We show that this fact can be easily deduced from the properties of the even integer part and of the even fractional part of the real number . Moreover, our further reasoning does not depend on how these two functions are defined on the odd integers.
We first prove that for arbitrary is an even rational number lying in . Observe that
| (10.67) |
and
| (10.69) |
yield that
| (A.27a) |
and, in view of (10.69),
| (A.27e) |
for implies that
| (A.27f) |
means that
and therefore . This property together with (A.27f) proves that is an even rational number lying in for arbitrary .
By applying the even integer and the even fractional parts to (A.27d), we obtain, taking into account of (A.27e),
Hence, we can easily calculate the coefficients of the continued fraction
by the formulas,
| (A.27g) |
and also,
| (A.27j) |
It follows from (A.27g) that
| (A.27k) |
We prove now that the following analogue of (A.27j) for an arbitrary even rational number , , , holds
| (A.27n) | |||
| (A.27o) |
where . Since for the integer is even, then with and , as follows from (A.27j) for . It remains to prove (A.27o) for . Since and are coprime then belongs to the set
and does not lie on the interval because . Thus, there exists a unique such that
and since , there also exists a unique satisfying
By setting and we obtain the existence of the integers and satisfying
| (A.27p) |
Here, the property (A.27o)(b) is immediate, while (A.27o)(a) follows from (jon, , p. 5, Lemma 1.5), which states that
| (A.27q) |
Finally, and yield and completes the proof of (A.27o).
We prove that an arbitrary even rational number , , , lying on the set , can be represented as with , , by employing induction on . It has already been written above that for the integer is even, and therefore with . Assume that and this statement is proved for all nonzero even rational numbers from the modulus of whose nominators is less than or equal . Let , , and . Invoking the induction hypothesis on the even rational number satisfying (A.27p), we obtain the existence of and such that . But then (A.27o) and (A.27p) yield
Lemma A.6
. Each nonzero even rational number in with nonzero coprime integers and can be uniquely represented in the form with , , satisfying
| (A.27r) |
Conversely, for arbitrary is an even rational number lying in . For different and from the numbers and are different.
28
We prove (5.95)(a). Since we can apply (5.15) to get . To prove the left-hand side equality
| (5.95(a)) |
while for every we can apply to (see (5.75))
to obtain
A.6 . Notes for Section 6
31
We prove (6.12). We introduce the parity indicator such that if the integer is even and if is odd. Let and be inverse to , i.e., for all . Denote . Then by (5.58) and (6.7),
Then
from which
where the right-hand side is if and (6.12) is proved for that case. But if then
because
This finished the proof of (6.12).
32
We prove (6.33). If then and
Therefore, in this case,
Let
we have , if . Together with (),
this yields the validity of (6.33) for ,
we have , and it follows from
| (A.32a) | ||||
that
while
which yields the validity of (6.33) for ,
This completes the proof of (6.33). Besides, we have proved that
| (A.32b) |
and
| (A.32c) |
A.7 . Notes for Section 7
33
We prove the left-hand side inequality of (7.1).
Since , and , , , it is enough to consider the case for which it is stated that
| (A.33a) |
We first assume that , , and prove that
| (A.33b) |
For any such that there exists such that , . Then it is necessary to obtain that
i.e.,
But on the interval the function increases from to . Therefore the above inequality is equivalent to
which is true because . Thus, (A.33b) is proved.
Then for such we deduce from the following consequence of (2.25),
and (A.33b) that
where by (ber1, , p. 325),
| (A.33c) |
It remains to prove (A.33a) for , , where due to -periodicity of it is possible to consider that . Then
and therefore
where by (ber1, , p. 325),
and the function has the derivative, satisfying
and therefore , . Thus,
where
and hence,
41
yields
where and therefore
Thus, by using , (see (bh1, , p. 35, (A.9c))), which actually gives , , we get
Applying , (see (bh1, , p. 35, (A.9d))), we deduce that
and therefore from the above inequality we obtain
So that
By using (3.20), we get
while
This proves
and completes the proof of (7.26).
43
For introduce
| (A.43b) |
where the contour passes from to .
By transforming the contour of integration in (A.43a) to and using Lemma 2.4, we obtain from the residue theorem (con, , p. 112) that
| (A.43e) |
This means that the function , being holomorphic on , coincides on the set with the function , which is holomorphic on .
By the uniqueness theorem for analytic functions (see (con, , p. 78)), we find that for , the analytic extension of the function from to (see (7.29)) equals the expression since holds for all , in view of (A.43e).
But by (5.79)(a) and (5.86), we see that , and since is simply connected it follows that the latter function is actually holomorphic on .
So that the equality
| (7.32) |
is proved for arbitrary and .
44
We prove (8.11). For , and the Poisson kernel
possesses the following property
i.e.,
| (A.44a) |
Since for every the Poisson kernel of the variable belongs to , by (1.6) and Jordan’s lemma (8.6), for every we have
from which
| (A.44b) |
Applying complex conjugation, taking into account that for arbitrary and , we get
| (A.44c) |
while
| (A.44d) |
Next,
i.e.,
| (A.44e) |
Applying complex conjugation, as above ,we get
| (A.44f) |
Finally,
| (A.44i) |
which completes the proof of (8.11).
A.8 . Notes for Section 8
References
- (1)
- (2) T. M. Apostol, Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976.338 pp.
- (3) Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi F., Higher transcendental functions. Vol. II. . McGraw-Hill Book Company, 1985.
- (4) G. A. Jones, J. M. Jones, Elementary number theory. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 1998. 301 pp
- (5) Prudnikov, A., Brychkov, Yu. and Marichev, O., Integrals and series, Vol. 1. Elementary functions. Gordon & Breach Science Publishers, New York, 1986.
- (6) Royden, H. L., Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.