Multivariate Lagrange interpolation and polynomials of one quaternionic variable

The quaternions $\mathbb{H}$ are a celebrated extension of the field of complex numbers to a noncommutative associative algebra over the real numbers (a skew-field) with elements

and (see [Rod14] for the basic theory of $\mathbb{H}$, which is assumed)

The Lagrange interpolant $Lf$ to a function $f$ at $n+1$ distinct points $x_{0},x_{1},\ldots,x_{n}$ in $\mathbb{R}$ or $\mathbb{C}$ is the unique polynomial of degree $n$ matching the values of $f$ at these points, which can be given explicitly as

Formally, the above formula makes sense for points in $\mathbb{H}$, giving an interpolant. However, due to the noncommutativity of the quaternions, the Lagrange polynomials depend on the order in which the product is evaluated. This is the first indication that the quaternionic polynomials of degree $n$ (as a right $\mathbb{H}$-module) might have a dimension greater than $n+1$ (depending on how they are defined). To resolve this impasse one could

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  Look for an interpolant from a fixed $(n+1)$-dimensional subspace of quaternionic polynomials of degree $n$.

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  Seek a “best” choice of Lagrange polynomials $\ell_{j}$ for given interpolation points, which would implicitly define an $(n+1)$-dimensional subspace of interpolants that is related to the geometry of the points.

The first approach has been considered by [Bol15], which we discuss in the next section. Our approach is the second. The essential features of each are (respectively):

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  Interpolation is not possible for all configurations of points. The condition for unique interpolation and the interpolation space are not translation invariant. The Lagrange polynomials $\ell_{j}$ may have zeros which are not interpolation points. It is possible to develop a Newton form and notion of divided difference.

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  Interpolation is possible for all configurations of points, and the interpolation space depends continuously on the points. The interpolant is translation invariant. The Lagrange polynomials $\ell_{j}$ can be chosen to be zero only at the interpolation points, and a Newton form can be developed.

We want polynomials and functions of a quaternionic variable to be an “$\mathbb{H}$-vector space”. To do this, we view such spaces a right $\mathbb{H}$-module (and co-opt the language of linear algebra). Linear maps (which are technically $\mathbb{H}$-homomorphisms) then act on the left, with the usual algebra of matrices then extending in the obvious way (cf. [Rod14]). The Lagrange interpolant, as defined above, is an $\mathbb{H}$-linear map, since

Bolotnikov [Bol15], [Bol20] uses the formal polynomials $\mathbb{H}[z]$ in $z$

on which a left and right evaluation at $a\in\mathbb{H}$ are defined by

For $\mathbb{H}[z]$ as right vector space, left evaluation is linear but right evaluation is not.

The (left) Lagrange interpolation of [Bol15] is from the $(n+1)$-dimensional subspace of polynomials of degree $n$ in $\mathbb{H}[z]$ to left evaluation at $n+1$ points in $\mathbb{H}$. Let us consider an example, to see the nature of this interpolation.

Two quaternions $q_{1}$ and $q_{2}$ are said to be similar (the term equivalent is used in [Bol15]) if $q_{2}=aq_{1}a^{-1}$ for some nonzero $a\in\mathbb{H}$. This is equivalent to $\mathop{\rm Re}\nolimits(q_{1})=\mathop{\rm Re}\nolimits(q_{2})$ and $|q_{1}|=|q_{2}|$. Hence $i,j,k$ are similar and $1$ and either of $j,k$ are not.

The left point evaluations $\delta_{a}:\mathbb{H}[z]\to\mathbb{H}:f\mapsto f^{e_{l}}(a)$, $a\in\mathbb{H}$, are $\mathbb{H}$-linear functionals on the right vector space $\mathbb{H}[z]$. These span a left vector space, with linear dependencies given by following:

To see this in play, we consider left quadratic Lagrange interpolation from $\mathbb{H}[z]$.

The Gauss elimination argument above shows that a necessary condition for left (or right) Lagrange interpolation by a polynomial in $\mathbb{H}[z]$ of degree $n$ to any $f$ to $n+1$ distinct points in $\mathbb{H}$ is that no three of the points are similar. This is in fact sufficient.

These results were developed from the notion of $P$-independence [Lam86], [LL88], i.e., the set of $n+1$ points $A$ is (left) $P$-independent (polynomial independent) if the linear functionals $\{\delta_{a}\}_{a\in A}$ above are linearly independent, or, equivalently, there is a subspace of $\mathbb{H}[z]$ of dimension $n+1$ from which unique (left) Lagrange interpolation is possible. This has recently been explored in the multivariate setting [MPnK19], [Mar20].

A symmetric form of (2.4) can be obtained by evaluating the symmetrised form of the linear dependence (2.3) at $f(z)=z^{2}$.

The condition for unique left Lagrange interpolation is not translation invariant, except for some real translations or interpolation to less than three points. For example, quadratic interpolation at $i,j,k$ is not possible, but it is possible at $2i,j+i,k+i$ (since $|2i|=2\neq\sqrt{2}=|j+i|$). In a similar vein, the polynomials of degree $n$ in $\mathbb{H}[z]$ when viewed as functions $f:\mathbb{H}\to\mathbb{H}:q\mapsto f_{0}+qf_{1}+\cdots+q^{n}f_{n}$ are not translation invariant, e.g.,

for some $b\in\mathbb{H}$, if and only if $a$ is real.

Since there is a unique one-dimensional subspace of polynomials of degree $n$ in $\mathbb{H}[z]$ whose left evaluation at $n$ points (with no three similar) is zero, a Newton form for (left) Lagrange interpolation can be developed.

It is now time to understand the nature of the quaternionic polynomials, viewed as functions $\mathbb{H}\to\mathbb{H}$, obtained from the formula (1.1) for the Lagrange polynomials. These involve polynomials of the form

which [Sud79] calls a quaternionic monomial of degree $r$. We define the $\mathbb{H}$-span of these monomials to be $\mathop{\rm Hom}\nolimits_{r}(\mathbb{H})$ the homogeneous polynomials of degree $r$, and $\mathop{\rm Pol}\nolimits_{n}(\mathbb{H})$ the polynomials of degree $k$ to be the $\mathbb{H}$-span of the homogeneous polynomials of degrees $\leq n$. These definitions extend to multivariate polynomials $\mathbb{H}^{d}\to\mathbb{H}$, where each occurrence of $q$ in the formula for a monomial is replaced by some coordinate $q_{j}$.

It is clear from the definitions, that the quaternionic polynomials are a graded ring, i.e., the product of homogeneous polynomials of degrees $j$ and $k$ is a homogeneous polynomial of degree $j+k$. To understand the dimensions of these spaces, we write $q\in\mathbb{H}$ as

and observe (see [Sud79]) that

Hence $t,x,y,z$ are homogeneous monomials (in $q$), as are $\overline{q}$ and $|q|^{2}=q\overline{q}$, i.e.,

Every monomial of degree $r$ can be written as a homogeneous polynomial of degree $r$ in the (real) variables $t,x,y,z$ with quaternionic coefficients. The monomials in $t,x,y,z$ are linearly independent over $\mathbb{H}$ by the usual argument (of taking Taylor coefficients), and so we have

In particular, the vector spaces of constant, linear and quadratic polynomials of a single quaternionic variable have dimensions $1,5$ and $15$, respectively. This contrasts sharply with the real and complex cases, where the dimensions are $1,2$ and $3$.

As is evident from this example, Lagrange interpolation from $\mathop{\rm Pol}\nolimits_{n}(\mathbb{H})$ is essentially interpolation from $\mathop{\rm Pol}\nolimits_{n}(\mathbb{R}^{4})$, with the additional feature that formulas can be developed using a single quaternionic variable $q$, or two complex variables $v,w$, where

We now consider interpolation from subspaces of $\mathop{\rm Pol}\nolimits_{n}(\mathbb{H})$ that are a quaternionic analogue of the holomorphic functions. A function $f:\mathbb{H}\to\mathbb{H}$ is said to be regular if it is in the kernel of the Cauchy-Feuter operator $2\partial_{\ell}$, i.e.,

and to be harmonic if

If $f$ is regular, then it is harmonic. The dimensions of $\mathop{\rm Reg}\nolimits_{n}(\mathbb{H})$ and $\mathop{\rm Harm}\nolimits_{n}(\mathbb{H})$, the regular and harmonic homogeneous polynomials of degree $n$, are

An explicit basis $\{P_{k\ell}^{n}-jP_{k-1,\ell}^{n}\}_{0\leq k\leq\ell\leq 1}$, for $\mathop{\rm Reg}\nolimits_{n}(\mathbb{H})$ is given in [Sud79], where

Here

so for $n=1$, $P_{00}^{1}(q)=v$, $P_{01}^{1}(q)=\overline{w}$, $P_{11}^{1}(q)=-\overline{v}$. But $Q_{01}^{1}(q)=\overline{w}$ is not regular. Interchanging $k$ and $\ell$ in the formula (presumably a typo), gives $P_{01}^{1}(q)=w$, and the basis

The product of regular functions is not (in general) regular, since

However, multiplying the basis of Example 3.2 by $i,j,k$, to get $it-x,jt-y,kt-z$, we have

so that the average ${1\over 2}\{(it-x)(jt-y)+(jt-y)(it-x)\}$ is regular. In this way, [Sud79] gives a basis for $\mathop{\rm Reg}\nolimits_{n}(\mathbb{H})$ consisting of symmetrised products of the linear regular polynomials $it-x,jt-y,kt-z$, where the factors occur $n_{1},n_{2},n_{3}$ times, $n_{1}+n_{2}+n_{3}=n$.

Though the (Feuter) regular polynomials are a proper subspace of the quaternionic polynomials (which share some aspects of holomorphic polynomials, but not that power series, since $q$ is not regular), we do not readily see a practical way to interpolate from them. There has recently been considerable interest in Cullen-regular functions [GS07], which do have power series expansions (around 0 or a real centre), and so correspond to the $(n+1)$-dimensional subspace of formal polynomials $\mathbb{H}[z]$ in $\mathop{\rm Pol}\nolimits_{n}(\mathbb{H})$, which we have already discussed.

We now consider interpolation methods derived from (1.1). As already discussed, this is essentially multivariate polynomial interpolation to functions $\mathbb{R}^{4}\to\mathbb{H}$.

Either of the choices for computing $\ell_{j}(x)$ suggested above give

• •

  A unique linear Lagrange interpolation operator to any points $x_{0},\ldots,x_{n}\in\mathbb{H}$ from the $(n+1)$-dimensional subspace $\mathop{\rm span}\nolimits\{\ell_{j}\}$ of $\mathop{\rm Pol}\nolimits_{n}(\mathbb{H})$, where the Lagrange polynomials have precisely $n$ zeros.

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  The operator depends continuously on the interpolation points.

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  It could be “symmetrised” to obtain an operator which doesn’t depend on the ordering of the points (though the Lagrange polynomials might now have additional zeros).

In this vein, we now define a generic polynomial of degree $n$ which is zero at $n$ points. For points $x_{1},\ldots,x_{n}\in\mathbb{H}$, let

where $S_{n}$ is the symmetric group. This polynomial of degree $n$ is zero at the points $x_{1},\ldots,x_{n}$, and (by construction) does not depend on their ordering.

We now present two possible choices for the Lagrange polynomials:

provided that $p(x_{j})\neq 0$, and

where the factors above are multiplied in the order $k=0,1,\ldots,n$ (or any fixed order). Both are independent of the point ordering, and depend continuously on the points. Let $L_{\Theta}$ be the corresponding Lagrange interpolation operator

for the points $\Theta=\{x_{0},\ldots,x_{n}\}\subset\mathbb{H}$, which does not depend on their ordering. We have

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  The interpolation operator $L_{\Theta}$ depends continuously on the points $\Theta$.

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  The interpolation space $\Pi_{\Theta}:=\mathop{\rm ran}\nolimits(L_{\Theta})$ depends continuously on the points $\Theta$.

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  The interpolation operator is translation invariant, i.e.,

  $$L_{\Theta+a}f(x)=L_{\Theta}\bigl{(}f(\cdot+a)\bigr{)}(x-a).$$

We compare this Lagrange interpolation with the two most prominent multivariate generalisations of univariate Lagrange interpolation, where the interpolation points are not in some predetermined geometric configuration.

Kergin interpolation [MM80] interpolates function values at $n+1$ points in $\mathbb{R}^{d}$ by a polynomial of degree $n$, with other “mean-value” interpolation conditions (see [Wal97]) that depend continuously on the points also matched. Here the interpolation space $\mathop{\rm Pol}\nolimits_{n}(\mathbb{R}^{d})$ is fixed, and hence depends continuously on the points. Kergin interpolation has also been extended to $\mathbb{C}^{d}$ (see [Fil97]). Using the identifications $\mathbb{H}\approx\mathbb{R}^{4}$, $\mathbb{H}\approx\mathbb{C}^{2}$ one can defined a Kergin interpolation to functions $\mathbb{H}\to\mathbb{H}$ from the whole space $\mathop{\rm Pol}\nolimits_{n}(\mathbb{H})$ (with additional interpolation conditions). The explicit formulas for Kergin interpolation involve derivatives of $f$ (the operator is defined for $C^{n}$-functions), and are not as easily computed as ours.

Least interpolation [dBR92] is a very general method, which seeks an interpolation space $\Pi_{\Theta}$ of dimension $n+1$ to the $n+1$ points $\Theta$, which has polynomials of lowest (least) degree. It has many nice properties that include the continuity properties listed above, but there is no explicit formula. It could be applied to functions $\mathbb{H}\to\mathbb{H}$ in the same way that Kergin interpolation can be. It might even be possible to develop a least interpolation for “polynomials in $\mathbb{H}^{d}$”, once such a theory is developed. The least interpolation has the advantage that polynomials of low degree are used. In particular, the Lagrange polynomials are a partition of unity, i.e.,

For the Lagrange polynomials that we have proposed, this may not be the case. Indeed, the set of all possible “nice” Lagrange polynomials $\ell_{j}$ for $x_{j}$, i.e, those polynomials $p=p_{\{x_{0},\ldots,x_{n}\}}$ of degree $n$, with $p(x_{k})=\delta_{jk}$ and $p$ not depending on the order of the points, form an affine subspace of $\mathop{\rm Pol}\nolimits_{n}(\mathbb{H})$, from which we have suggested two choices. It may be that requiring, in addition, the partition of unity property, gives a unique choice, but we have not pursued this.

Since our Lagrange interpolation is effectively interpolation to functions $\mathbb{R}^{4}\to\mathbb{H}$, we can define an interpolation operator to functions $\mathbb{R}^{4}\to\mathbb{R}$ in the natural way, i.e.,

and $\Theta$ is the points viewed as a subset of $\mathbb{H}$. Heuristically, the real Lagrange polynomials $\hat{\ell}_{j}=\mathop{\rm Re}\nolimits(\ell_{j})$ are more likely to be “nice”, e.g., form a partition of unity, have no extra zeros, or coincide for both choices, since there is the “commutativity relation”

In a similar way, one could define a Lagrange interpolation operator to functions $\mathbb{C}^{2}\to\mathbb{C}$, by using the Cayley-Dickson construction (3.11).

It is also possible to develop Lagrange interpolants through a Newton form (either for functions $\mathbb{H}\to\mathbb{H}$ or $\mathbb{R}^{4}\to\mathbb{R}$), and an associated theory of divided differences. For example, if $L_{n-1}f$ is a Lagrange interpolant at $x_{1},\ldots,x_{n-1}\in\mathbb{H}$, then

gives a Lagrange interpolant $L_{n}f$ to $f$ at the points $x_{0},\ldots,x_{n}$, and a “divided difference” $[x_{0},\ldots,x_{n}]f\in\mathbb{H}$. Choices for $p$ could include $p_{\{x_{0},\ldots,x_{n-1}\}}$ or $\ell_{n}$ (for $x_{0},\ldots,x_{n}$). The map $f\mapsto[x_{0},\ldots,x_{n}]f\in\mathbb{H}$ is an $\mathbb{H}$-linear functional. We have not invesigated its divided difference type properties any further.

This work came about when investigating tight frames and spherical designs for $\mathbb{H}^{d}$ (see [Wal18], [Wal20]). It soon became apparent that the theory of quaternionic polynomials of one and several variables is involved, and not widely known. There is considerable work in at least two different directions. One is a formal (algebraic) approach dating back to [Ore33], [Lam86], where point evaluation and multiplication of polynomials are appropriately defined, and the other views the polynomials as functions, with the usual point evaluation and (pointwise) multiplication.

Here we have shown that

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  The space of quaternionic polynomials required for the classical formula (1.1) for Lagrange interpolation to make sense has a high dimension (3.10).

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  Lagrange interpolation methods with some geometric properties, e.g., translation invariance, are essentially multivariate polynomial interpolation methods.

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  Many basic questions about quaternionic polynomials of one (and several) variables remain, e.g., the existence of Lagrange polynomials with no extra zeros which form a partition of unity.

Functions of quaternionic variables have long been used in physics, and geometric design (cf. [FGMS19]). We hope this paper gives some insight into polynomials of one or more quaternionic variables, and their use in interpolation and cubature (spherical designs).