Journal article 1760 views 305 downloads
An infinite dimensional umbral calculus
,
Yuri Kondratiev,
Eugene Lytvynov ,
Maria João Oliveira
Journal of Functional Analysis, Volume: 276, Issue: 12, Pages: 3714 - 3766
Swansea University Authors:
Dmitri Finkelshtein , Eugene Lytvynov
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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...
| Published in: | Journal of Functional Analysis |
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| ISSN: | 00221236 |
| Published: |
2019
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| Online Access: |
Check full text
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa49896 |
| 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. |
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| Keywords: |
Polynomial sequence of binomial type on $\mathcal D'$; Sheffer sequence on $\mathcal D'$; shift-invariant operators; umbral calculus on $\mathcal D'$ |
| College: |
Faculty of Science and Engineering |
| Issue: |
12 |
| Start Page: |
3714 |
| End Page: |
3766 |