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Fourier–Laplace transforms and orthogonal polynomials

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Abstract

The holomorphic extensions Fα(iz), zd, of the Fourier transforms Fαf̂α of a sequence of continuous real functions fα(x), xRd,αN0d, with exponential decay at infinity, generates a sequence of polynomials Qβ, βN0d, of degree |β|, that are biorthogonal to the distributional derivatives μα(1)|α|fα(α), αN0d. The generating function is a generalized Taylor series expansion of the Fourier-Laplace kernel exz in terms of μ̂α(iz)=zαFα(iz). Sufficient conditions on Fα(iz) are given for the series expansion of holomorphic functions in terms of zαFα(iz), αN0d, from which the generalized generating function is derived. In the case μα=ωPα, where ω is a weight function and Pα is a polynomial of degree |α|, Qα is a constant multiple of Pα and for n=1,2,, the set {Qα:αN0d,|α|=n}, suitably normalized, is an orthonormal basis for the space of orthogonal polynomials of degree n associated with Pα.

Introduction

The Gaussian function G(x)1(2π)dexx2, xRd, plays an important role in many fields of Mathematics. Its Fourier transform Ĝ(u), uRd, which is also a Gaussian, can be extended to an entire function on the complex Euclidean space d{z=(z1,z2,,zd):zj,j=1,2,,d}, via its Fourier-Laplace transform Ĝ(iz)RdexzG(x)dx=ezz2, zd. On the other hand, the power series expansion exzĜ(iz)=αN0Hα(x)zα,zd,generates the sequence of Hermite polynomials Hα, αN0d, where N0d is the set of d-dimensional vectors with nonnegative integer components. The generating function (1.1) can be expressed in the form exz=αN0Hα(x)GHα̂(iz),zd,since zαĜ(iz)=(1)|α|G(α)̂(iz)=GHα̂(iz), which is the Rodrigues formula for the tensor product Hermite polynomials in the Fourier–Laplace transform domain. Restricting z=iu, uRd,(1.2) gives a relationship, which expresses the Fourier transform kernel eixu as a series expansion in terms of the Hermite polynomials Hα(x) and the Fourier transforms GHα̂(u) of the weighted Hermite polynomials GHα. Note that the Hermite polynomials Hα and the weighted polynomials GHα are a biorthogonal pair, and (1.2) exhibits a symbiotic relationship between them. The objective of this note is to extend this relationship to a wider class of orthogonal and biorthogonal polynomials. This is a multivariate version of the generalized generating functions for the construction of biorthogonal pairs of polynomials and spline functions that extent orthogonal polynomials in a natural way [4]. In this paper the focus is on the general multivariate theory and specific applications to some classical multivariate orthogonal polynomials. The general theory goes beyond orthogonal polynomials and provides an extension of the connections between orthogonal polynomials and probability distributions to biorthogonal polynomials and multivariate spline functions and joint probability distributions of random vectors.

We shall use the following standard multivariate notations. For ζ=(ζ1,ζ2,,ζd),z=(z1,z2,,zd)d, α=(α1,α2,,αd),β=(β1,β2,,βd)N0d, |α|α1+α2+αd, zαz1α1z2α2zdαd, ζzζ1z1+ζ2z2++ζdzd, |z|={|z1|2+|z2|2++|zd|2}12, zζ(z1ζ1,z2ζ2,,zdζd), and if ζj0,j=1,2,,d, zζ(z1ζ1,z2ζ2,,zdζd). For a nonnegative integer n, let Znd{αN0d:|α|=n}, and so N0d=n=0Znd, a disjoint union. It is wellknown that the lexicographic ordering on N0d is a linear order. Here we use a slightly modified form of the lexicographic ordering, which is called the graded lexicographic ordering on N0d defined by α<β if and only if |α|<|β| or |α|=|β|, αj=βj, j=1,2,,k1 for some kd, and αk<βk. The graded lexicographic ordering is lexicographic on each subset Znd, n=0,1,, of N0d. Since these subsets form a partition of N0d, and αZd is strictly less than βZnd for <n, it follows that the graded lexicographic ordering is also a linear ordering. For α,βN0d, we shall also write αβ to mean αjβj, for all j=1,2,d, and αβ if βjαj<0, for some j. If αβ, we shall say α precedes β. Let Nd be the set of all elements in N0d that are linearly order by , i.e. for all α,βNd, αβ or βα, and NdN0dNd. Note that α if and only if =α+β for some βN0d. For zd and αN0d, the differential operator zα|α|z1α1zdαd, and f(α)(z)zαf(z). For n=0,1,, let Pndspanxα:αN0d,|α|=n be the space of homogeneous polynomials of degree n and Πndspanxα:|α|n,αN0d be the space of all polynomials of degree n. Note that Πnd=PndPn1dP0d, a direct sum of the space of homogeneous polynomials, and the dimension of Pnd=d+n1n and that of Πnd=d+nn. A useful reference on multivariate orthogonal polynomials can be found in [3].

In the next section we describe the general setting and state the main theorem. The main result is a relationship that expresses the Fourier–Laplace kernel exz as a series expansion in terms of a sequence of multivariate polynomials Qα(x), xRd,αN0d, and the Fourier–Laplace transforms μ̂α(iz) of a sequence of functions (more generally, tempered distributions) μα, in the same way as (1.2) expresses its relationship with the Hermite polynomials Hα(x) and the Fourier–Laplace transformation GHα̂(iz) of the weighted Hermite polynomials GHα. This relationship also provides a generalized generating function for the polynomials Qα, from which their properties can be derived. Whereas the classical generating functions for orthogonal polynomials are power series expansions, which are available only to a limited class of orthogonal polynomials such as Hermite, Laguerre and Legendre polynomials, the generalized generating functions are generalized Taylor expansions in terms of μ̂α(iz), which are available to a wider class of orthogonal polynomials as well as biorthogonal polynomials.

The main theorem and its proof are given in Section 2. The main results in this section and in Section 3 are on the expansions of holomorphic functions in several complex variables in terms of the sequence μ̂α(iz), αN0d, which are a generalization of the Taylor series expansions. This is generally a difficult problem, even in the case of one complex variable, which has a long history, but few results [1], [2], [5]. Section 3 gives another sufficient condition for the generalized Taylor expansion of holomorphic functions, which supplement that in Section 2. It is shown in Corollary 3.7 that if μα=ωPα, αN0d, where ω is a weight function and Pα is a sequence of polynomials of exact degree |α|, then for each α, Qα is a constant multiple of Pα and for n=1,2,, the set {Qα:αN0d,|α|=n}, suitably normalized, is an orthonormal basis for the space of orthogonal polynomials of degree n associated with Pα. The general theory developed in Sections 2 The general setting and main theorem, 3 The second generalized generating functions includes the well-known classical orthogonal polynomials such as Hermite, Laguerre and Jacobi polynomials on the simplex, which are considered in Section 4.

Section snippets

The general setting and main theorem

In this section we develop the general setting, state and prove the main theorem and other related results. Let fα, αN0d, be a sequence of continuous functions on Rd that satisfy the condition |fα(x)|Kαej=1d(Bj+ϵ)|xj| as |xj|,j=1,2,,d,for any arbitrary small ϵ>0 and some B=(B1,B2,,Bd)R+d, which are independent of α, and constant Kα, which may depend on α. Then fαL1(R)L2(Rd) and their Fourier transforms Fαf̂α can be extended to holomorphic functions Fα(iz)Rdezxfα(x)dx,zSB,where SB

The second generalized generating functions

Theorem 2.10 gives a sufficient condition, i.e. the uniform boundedness of the Fourier–Laplace transforms Fα(iz) in a neighbourhood of the origin, for the generalized Taylor series expansion of holomorphic functions in terms zαFα(iz), αN0d. In order to extend the general theory to cover a wider class of orthogonal and biorthogonal polynomials we shall consider another sufficient condition on Fα(iz) for such an expansion, in which |Fα(iz)| need not be uniformly bounded and the exponential decay

Classical orthogonal polynomials

The general theory developed in the previous sections includes the classical multivariate Hermite, Laguerre and Jacobi polynomials, which shall be briefly described here.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The proof of Proposition 2.7, which is a crucial auxiliary result in the proof of the main theorem, was suggested by an anonymous referee. It is elegantly shorter and delivers a sharper bound in (2.19) than the original version of the proof by the author. Thanks to the referee for the suggestion and other related comments that help to improve the presentation of the paper. Thanks also to Professor Goh Say Song for useful discussions and to the Department of Mathematics, National University of

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