Full Length ArticleFourier–Laplace transforms and orthogonal polynomials
Introduction
The Gaussian function , , plays an important role in many fields of Mathematics. Its Fourier transform , , which is also a Gaussian, can be extended to an entire function on the complex Euclidean space , via its Fourier-Laplace transform , . On the other hand, the power series expansion generates the sequence of Hermite polynomials , , where is the set of -dimensional vectors with nonnegative integer components. The generating function (1.1) can be expressed in the form since , which is the Rodrigues formula for the tensor product Hermite polynomials in the Fourier–Laplace transform domain. Restricting , ,(1.2) gives a relationship, which expresses the Fourier transform kernel as a series expansion in terms of the Hermite polynomials and the Fourier transforms of the weighted Hermite polynomials . Note that the Hermite polynomials and the weighted polynomials 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 , , , , , , , and if , . For a nonnegative integer , let , and so , a disjoint union. It is wellknown that the lexicographic ordering on is a linear order. Here we use a slightly modified form of the lexicographic ordering, which is called the graded lexicographic ordering on defined by if and only if or , , for some , and . The graded lexicographic ordering is lexicographic on each subset , , of . Since these subsets form a partition of , and is strictly less than for , it follows that the graded lexicographic ordering is also a linear ordering. For , we shall also write to mean , for all , and if , for some . If , we shall say precedes . Let be the set of all elements in that are linearly order by , i.e. for all , or , and . Note that if and only if for some . For and , the differential operator , and . For , let be the space of homogeneous polynomials of degree and be the space of all polynomials of degree . Note that , a direct sum of the space of homogeneous polynomials, and the dimension of and that of . 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 as a series expansion in terms of a sequence of multivariate polynomials , , and the Fourier–Laplace transforms of a sequence of functions (more generally, tempered distributions) , in the same way as (1.2) expresses its relationship with the Hermite polynomials and the Fourier–Laplace transformation of the weighted Hermite polynomials . This relationship also provides a generalized generating function for the polynomials , 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 , 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 , , 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 , , where is a weight function and is a sequence of polynomials of exact degree , then for each , is a constant multiple of and for , the set , suitably normalized, is an orthonormal basis for the space of orthogonal polynomials of degree associated with . 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 , , be a sequence of continuous functions on that satisfy the condition for any arbitrary small and some , which are independent of , and constant , which may depend on . Then and their Fourier transforms can be extended to holomorphic functions where
The second generalized generating functions
Theorem 2.10 gives a sufficient condition, i.e. the uniform boundedness of the Fourier–Laplace transforms in a neighbourhood of the origin, for the generalized Taylor series expansion of holomorphic functions in terms , . In order to extend the general theory to cover a wider class of orthogonal and biorthogonal polynomials we shall consider another sufficient condition on for such an expansion, in which 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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