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An infinite dimensional umbral calculus

Dmitri Finkelshtein Orcid Logo, Yuri Kondratiev, Eugene Lytvynov Orcid Logo, Maria João Oliveira

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

Swansea University Authors: Dmitri Finkelshtein Orcid Logo, Eugene Lytvynov Orcid Logo

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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...

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Published in: Journal of Functional Analysis
ISSN: 00221236
Published: 2019
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URI: https://cronfa.swan.ac.uk/Record/cronfa49896
first_indexed 2019-04-04T16:40:52Z
last_indexed 2020-07-10T03:11:26Z
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spelling 2020-07-09T20:37:34.5939892 v2 49896 2019-04-04 An infinite dimensional umbral calculus 4dc251ebcd7a89a15b71c846cd0ddaaf 0000-0001-7136-9399 Dmitri Finkelshtein Dmitri Finkelshtein true false e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2019-04-04 MACS 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. Journal Article Journal of Functional Analysis 276 12 3714 3766 00221236 Polynomial sequence of binomial type on $\mathcal D&apos;$; Sheffer sequence on $\mathcal D&apos;$; shift-invariant operators; umbral calculus on $\mathcal D&apos;$ 16 6 2019 2019-06-16 10.1016/j.jfa.2019.03.006 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2020-07-09T20:37:34.5939892 2019-04-04T14:06:47.1430334 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Dmitri Finkelshtein 0000-0001-7136-9399 1 Yuri Kondratiev 2 Eugene Lytvynov 0000-0001-9685-7727 3 Maria João Oliveira 4 0049896-04042019140737.pdf Finalversionv3.pdf 2019-04-04T14:07:37.9530000 Output 418786 application/pdf Accepted Manuscript true 2020-04-01T00:00:00.0000000 Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND). true eng
title An infinite dimensional umbral calculus
spellingShingle An infinite dimensional umbral calculus
Dmitri Finkelshtein
Eugene Lytvynov
title_short An infinite dimensional umbral calculus
title_full An infinite dimensional umbral calculus
title_fullStr An infinite dimensional umbral calculus
title_full_unstemmed An infinite dimensional umbral calculus
title_sort An infinite dimensional umbral calculus
author_id_str_mv 4dc251ebcd7a89a15b71c846cd0ddaaf
e5b4fef159d90a480b1961cef89a17b7
author_id_fullname_str_mv 4dc251ebcd7a89a15b71c846cd0ddaaf_***_Dmitri Finkelshtein
e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov
author Dmitri Finkelshtein
Eugene Lytvynov
author2 Dmitri Finkelshtein
Yuri Kondratiev
Eugene Lytvynov
Maria João Oliveira
format Journal article
container_title Journal of Functional Analysis
container_volume 276
container_issue 12
container_start_page 3714
publishDate 2019
institution Swansea University
issn 00221236
doi_str_mv 10.1016/j.jfa.2019.03.006
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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.
published_date 2019-06-16T07:29:06Z
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