Distributional Weight Functions for Orthogonal Polynomials

Abstract

Given any collection of real numbers $\{ \mu _i \} _{i = 0}^\infty $, called moments, satisfying a Hamburger-like condition $\Delta _n = \det [\mu _{i + j} ]_{i,j = 0}^n \ne 0$ and a growth condition $| {\mu _n } | < cM^n n!$, where c, M are constant, $n = 0,1, \cdots $, the Chebyshev polynomials $p_0 = 1$, \[p_n (x) = \left[ {{1 / {\Delta _{n - 1} }}} \right]\left| {\begin{array}{*{20}c} {\mu _0 } & {\mu _1 } & \cdots & {\mu _n } \\ \vdots & {} & {} & \vdots \\ {\mu _{n - 1} } & {\mu _n } & \cdots & {\mu _{2n - 1} } \\ 1 & x & \cdots & {x^n } \\ \end{array} } \right|,\]$n = 1,2, \cdots $ are shown to be orthogonal with respect to the linear functional \[w(x) = \sum_{n = 0}^\infty {( - 1)^n \mu _n \delta ^{(n)} } {{(x)} / {n!}}.\] The problem of the existence of extensions of w to a space of test functions which includes polynomials is also discussed. It is shown that if $F^{ - 1} w(t)$ has an analytic continuation which has a classical Fourier transform, then that transform is the desired extension. If the continuation has an appropriate derivative which has a classical Fourier transform, then there exists a canonical regularizarion of a regular distribution which extends w. As examples the Legendre, Jacobi, Laguerre, generalized Laguerre, Hermite and Bessel polynomials are offered. The Fourier transform establishes the connection between the functionals w and the classical weight functions when they exist. Further an extension of classical results is made in the cases of the generalized Laguerre and Jacobi polynomials. In the case of the Bessel polynomials, however, the measure of bounded variation, guaranteed by Boas’s theorem, can only be found (?) as a Fourier transform, and so still remains an enigma.