The system of translates and the special affine Fourier transform

The special affine Fourier transform (SAFT) was first considered by S. Abe and J. T. Sheridan in [1] for the study of certain operations on optical wave functions. The SAFT is formally defined as

where $A$ stands for the set $\{a,b,c,d,p,q\}$ of real parameters which satisfy the relation $ad-bc=1$. The integral transform (1.1) is related to the special affine linear transform of the phase space

Due to the conditions on the parameters $a,b,c,d$ the matrix in (1.2) belongs to the special linear group $\mathrm{SL}(2,\mathbb{R})$ and the affine transform (1.2) is therefore given by elements from the inhomogeneous linear group

This justifies the name special affine Fourier transform for (1.1). The action of the group $\mathrm{SL}(2,\mathbb{R})$ on the time frequency plane and the relation to quadratic Fourier transforms is well studied. We will not go into the details here but refer to the book [11].

A number of important transforms are special cases of the SAFT. For example, $A=\{0,1,-1,0,0,0\}$ gives the ordinary Fourier transform and $A=\{0,-1,1,0,0,0\}$ its inverse. The parameter set $A=\{\cos\theta,\sin\theta,-\sin\theta,\cos\theta,0,0\}$ gives the fractional Fourier transform, and $A=\{1,\lambda,0,1,0,0\}$ produces the Fresnel transform.

In optics, certain one parameter subgroups of the $\mathrm{ISL}(2,\mathbb{R})$ are of special interest. Among them are the fractional Fourier transform, the Fresnel transform (also called free space propagation in this context), the hyperbolic transform $A=\{\cosh\theta,\sinh\theta,\sinh\theta,\cosh\theta,0,0\}$, the lens transform $A=\{1,0,\lambda,1,0,0\}$, and the magnification transform $A=\{e^{\beta},0,0,e^{-\beta},0,0\}$. The latter two cases need a careful analysis for the limit case $b\to 0$ which we will not consider in this paper. We will not try to expound the various connections to optics further but refer to [15] for more details.

In this paper, we consider (1.1) from the point of view of applied harmonic analysis and take it as a signal transform of a (suitable) function. We are mainly interested in studying the principal shift invariant spaces and sampling theorems related to the SAFT. In the classical case, the principal shift invariant space generated by $\phi\in L^{2}(\mathbb{R})$ is defined as $V(\phi)=\overline{span}\{T_{k}\phi=\phi(\cdot-k):k\in\mathbb{Z}\}$. The classical Fourier transform ($A=\{0,1,-1,0,0,0\}$ in (1.1)) plays a crucial role for the analysis of such spaces. The crucial point is that the ordinary translation and the classical Fourier transform are intimately related. This is due to the identity $e^{i\omega(t-x)}=e^{i\omega t}e^{-i\omega x}$ which gives, as a consequence the convolution theorem, the relation between translation, modulation, Fourier transform etc. These theorems are used over and over again in Fourier analysis and in the study of shift invariant spaces in particular. It is completely obvious that the ordinary translation does not interact nicely with the SAFT (resp. its complex exponential kernel). Hence working with the SAFT a new concept of a translation is needed. This generalized translation should be linked to the SAFT in an analogous manner as the ordinary translation is linked to the classical Fourier transform. If this is the case, then it seems reasonable to expect that the central theorems (convolution theorem etc.) hold in a similar manner. An idea for the construction of such generalized translation comes from the observation that in the classical setting we have

In this paper, we define a new translation operator $T^{A}_{x}$, which serves our purpose in the case of SAFT.

In [6] A. Bhandari and A.I. Zayed considered chirp-modulation and used this to obtain a convolution theorem. However, they did not define a generalized translation operator explicitly, and hence did not investigate the consequences of this concept with respect to harmonic analysis. We shall demonstrate that the generalized translation is the suitable concept to obtain analogues of fundamental theorems such as Wendel’s theorem for the multipliers, Wiener theorem and Weiner-Tauberian theorem in connection with the closed ideals of translation invariant spaces in the context of the SAFT. For a study of multipliers and Wendel’s theorem for the Fourier transform we refer to [18], for Wiener theorem and Wiener-Tauberian theorem we refer to [12] and [22]. The novelty of this approach is that apart from these theorems, one can look into the study of multiplier theory, including Hörmander multiplier theorem in the SAFT domain. Moreover, one can define an appropriate modulation operator in connection with the SAFT, using which one can define modulation spaces associated with the SAFT. This in turn, motivates to study multiplier results for the new modulation spaces. (See [7] in this connection.)

Using the new translation operator $T^{A}_{x}$ the $A$-shift invariant spaces are defined as $V_{A}(\phi)=\overline{span}\{T^{A}_{k}\phi:k\in\mathbb{Z}\}$ for an appropriate function $\phi\in L^{2}(\mathbb{R})$. When $\phi$ belongs to the Wiener amalgam space $W\big{(}C,\ell^{1}(\mathbb{Z})\big{)}$ the space $V_{A}(\phi)$ turns out to be a reproducing kernel Hilbert space. Moreover, we give characterization theorems for the system of translates $\{T^{A}_{k}\phi:k\in\mathbb{Z}\}$ to be a frame sequence, orthogonal system or a Riesz sequence. If the system of translates is a frame then an important question is about the nature of the dual frame elements. We show that in our setting, the elements of the dual frame of system of $A$-translates are also $A$-translates of a single function. For a study of shift invariant spaces, system of translates, frames and Riesz basis in the classical case, we refer to Christensen [10].

In the final part of the paper we study the sampling in $A$-shift invariant spaces. A fundamental problem in sampling theory is to find, for a certain class of functions, appropriate conditions on a countable sampling set $X=\{x_{j}\in\mathbb{R}:j\in J\}$ under which a given function $f\in V$ can be reconstructed uniquely and stably from the samples $\{f(x_{j}):j\in J\}$. We refer to the work of Butzer and Stens [8] for a review on sampling theory and its history. When $V$ is the classical principal shift invariant space with a single generator $V(\phi)$ or multi-generators $V(\phi_{1},\phi_{2},...,\phi_{r}),~{}\phi,\phi_{1},\phi_{2},...\phi_{r}\in L^{2}(\mathbb{R})$, there is a huge literature available on several interesting problems connected with sampling theory starting from the fundamental Shannon sampling theorem. We cite only a few references in this connection for the reader to get familiarity with this subject matter. (See [2], [3], [4], [5], [9], [13], [14], [17], [19], [20], [24], [25], [26], [27]).

In this paper, similar to Theorem $4.2$ of [3] and Theorem $2.1$ of [23], we obtain equivalent conditions for a set $X$ to be a stable set of sampling for $V_{A}(\phi)$ in terms of the operator $U$ where $U_{jk}=T_{k}^{A}\phi(x_{j})$, the reproducing kernel and the Zak transform $Z_{A}$, which we introduce here. We also obtain a sufficient condition for the set of integers to be a stable set of sampling for $V_{A}(\phi).$ In the study of non uniform sampling and average sampling, Bernstein type inequalities play an important role. In this paper we obtain an analogue of Bernstein type inequality for $V_{A}(\phi)$. However, we do not intend to study non-uniform sampling and average sampling in this paper. We focus on uniform sampling. In particular, when $\mathbb{Z}$ turns out to be a stable set of sampling, we obtain a reconstruction formula and hence a sampling theorem in the sense of $L^{2}$ convergence for certain $A$-shift invariant spaces $V_{A}(\phi)$. Further, under some additional hypotheses on $\phi$, we obtain a sampling formula in the sense $L^{2}$ convergence and uniform convergence. As corollaries we obtain Shannon sampling theorem in the SAFT domain and sampling theorem for the $A$-shift invariant space generated by second order $B$-spline.

We organize our paper as follows. In section $2$, we define $A$-convolution of a measure and a function and chirp modulation $C_{s}\mu$ of a measure $\mu$. Using this and the function $\rho_{A}(x)=e^{\frac{i}{2b}(ax^{2}+2px)}$, we obtain a relation between classical translation and $A$-translation. We prove an analogue of Wendel’s theorem for the SAFT. In section $3$, we study closed ideals in the Banach algebra $(L^{1}(\mathbb{R}),\star_{A})$. We obtain analogues of Wiener theorem and Wiener-Tauberian theorem in the context of the SAFT. In section $4$, we study $A$-shift invariant spaces and their theoretical aspects. Section $5$ as well as Section $6$ are devoted to sampling theorems in $A$-shift invariant spaces. Finally in Section $7$, we present a local reconstruction method for sampling in $A$-shift invariant spaces along with implementation.

Now we shall provide the necessary terminology and background for this paper.

Let $0\neq\mathcal{H}$ be a separable Hilbert space.

The numbers $m,M$ are called frame bounds. If we have the right hand side inequality for a sequence in $\mathcal{H}$, then that sequence is called a Bessel sequence.

It is well known that a sequence $\{f_{k}:k\in\mathbb{N}\}$ is a Riesz sequence if there exist $m,M>0$ such that its Gramian $G$ satisfy the following inequality:

In this section, we introduce $A$-translation operator in connection with the SAFT. Using $A$-translation operator, we define $A$-convolution of a regular Borel measure and a function. Further, we obtain an analogue of Wendel’s theorem. Towards this end, first we extend the definition of SAFT to the space of all regular Borel measures.

With the help of the following auxiliary functions

the SAFT can be expressed as

Since $|\eta_{A}(\omega)|=1=|\rho_{A}(t)|$ for all $t,\omega\in\mathbb{R}$, we immediately see from (2.4) that $\mathscr{F}_{A}f\in C_{0}(\mathbb{R})$ with $\|\mathscr{F}_{A}f\|_{\infty}\leq(2\pi|b|)^{-1/2}\,\|f\|_{1}$. Moreover, (2.4) also shows that $\mathscr{F}_{A}$ can be extended to $L^{2}(\mathbb{R})$ and defines a unitary operator on that space. In particular,

The inverse of $\mathscr{F}_{A}$ on $L^{2}(\mathbb{R})$ can also be easily determined using (2.4)

Finally, (2.4) also provides an extension of the SAFT to $M(\mathbb{R})$, the space of all complex valued bounded regular Borel measures on $\mathbb{R}$, equipped with the total variation norm. For $\mu\in M(\mathbb{R})$, we have

with $\mathscr{F}_{A}\mu\in C_{b}(\mathbb{R})$ and $\|\mathscr{F}_{A}\mu\|_{\infty}\leq(2\pi|b|)^{-1/2}\|\mu\|_{M(\mathbb{R})}$.

We now introduce the generalized translation operator associated with SAFT. In order to do so, we fix the following notation

It is easy to see that $T_{x}^{A}$ is norm preserving in all spaces $L^{p}(\mathbb{R}),~{}1\leq p\leq\infty$ or $C_{0}(\mathbb{R})$.

We can relate our new translation and the classical translation in the following way, using $\rho_{A}$ and the chirp modulation operator $C_{\frac{a}{b}},$ where

In fact,

Similarly one can show (2.9).

Now, we collect the properties of $T_{x}^{A}.$

It is interesting to note that the map $x\mapsto T_{x}$ is a group representation, whereas from Proposition 2.3 (i) it follows that $x\mapsto T_{x}^{A}$ is just a projective representation in general, which shows that the new translation is fundamentally different from that of the classical translation. Proposition 2.3 (iii) is what is known in harmonic analysis as a product formula. The relation (2.10) extends to functions $f\in L^{2}(\mathbb{R})$ as well. For those functions we have in particular

where the equality holds in the sense of $L^{2}(\mathbb{R})$ functions.

Clearly, $C_{s}\mu\in M(\mathbb{R}).$ Similarly, we can define $\rho_{A}\mu$ as $d(\rho_{A}\mu)(x)=\rho_{A}(x)d\mu(x)$.

Using the $A$-translation, we define the $A$-convolution of $\mu\in M(\mathbb{R})$ and $f\in L^{1}(\mathbb{R})$ as

The integral in (2.9) can also be viewed as a vector-valued integral as follows.

where the right hand side is a Bochner integral. The convergence of the integral follows from

Now we give a relation between classical convolution and $A$-convolution of a measure and a function. Consider

using (2.8), which in turn implies that

Further, using (2.9), one can show that

The convolution theorem for the SAFT reads as follows.

In particular, if $\mu=g\,dt,$ for some $g\in L^{1}(\mathbb{R})$ then

See [6] for more details.

The concept of chirp modulation was used in [6] to define a chirp convolution and to get sampling theorems. Although the $A$-translation is somehow included implicitly in the definition of the chirp convolution, it has not been used to its full extent in [6] and hence the harmonic analysis of the special affine Fourier transform has not been developed. However we want to make use of our new translation from harmonic analysis point of view. Towards this end we first we prove an analogue of Wendel’s theorem for the SAFT.

We are now in a position to state one of our main results. The following statement is an analogue of Wendel’s theorem in the context of the SAFT.

We end this section by establishing an analogue of the Poisson summation formula for the SAFT and the corresponding $A$-translation. From now on, we use the following notation.

We refer to [6] for the proof.

We have seen that

Thus it is easy to see that $\|g\star_{A}f\|_{1}\leq(2\pi|b|)^{-1/2}\|g\|_{1}\,\|f\|_{1}$, from which it follows that $(L^{1}(\mathbb{R}),\star_{A})$ is a commutative Banach algebra. In this section, we aim to study the closed ideals in $(L^{1}(\mathbb{R}),\star_{A})$. Towards this end, first we show that $\big{(}L^{1}(\mathbb{R}),\star_{A}\big{)}$ possesses a bounded approximate identity as in the classical case $(L^{1}(\mathbb{R}),\star)$.

Now, we aim to study $A$-translation invariant closed ideals in $(L^{1}(\mathbb{R}),\star_{A})$. First we prove the following theorem in this context.

Now we are in a position to state and prove an analogue Weiner’s theorem in connection with the SAFT.