### An infinite dimensional umbral calculus / Dmitri Finkelshtein, Yuri Kondratiev, Eugene Lytvynov, Maria João Oliveira

Journal of Functional Analysis, Volume: 276, Issue: 12, Pages: 3714 - 3766

Swansea University Authors:

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Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND).

DOI (Published version): 10.1016/j.jfa.2019.03.006

Abstract

The aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial s...

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Published in: Journal of Functional Analysis 00221236 2019 https://cronfa.swan.ac.uk/Record/cronfa49896 No Tags, Be the first to tag this record!
Abstract: The aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on $\mathbb R$ of binomial type to a polynomial sequence of binomial type on $\mathcal D'$, and a lifting of a Sheffer sequence on $\mathbb R$ to a Sheffer sequence on $\mathcal D'$. Examples of lifted polynomial sequences include the falling and rising factorials on $\mathcal D'$, Abel, Hermite, Charlier, and Laguerre polynomials on $\mathcal D'$. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role. Polynomial sequence of binomial type on $\mathcal D'$; Sheffer sequence on $\mathcal D'$; shift-invariant operators; umbral calculus on $\mathcal D'$ College of Science 12 3714 3766