Journal article 994 views 171 downloads
An infinite dimensional umbral calculus
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 |
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2019
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URI: | https://cronfa.swan.ac.uk/Record/cronfa49896 |
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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 SMA 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 $\mathcal D'$; Sheffer sequence on $\mathcal D'$; shift-invariant operators; umbral calculus on $\mathcal D'$ 16 6 2019 2019-06-16 10.1016/j.jfa.2019.03.006 COLLEGE NANME Mathematics COLLEGE CODE SMA 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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facultyofscienceandengineering |
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Faculty of Science and Engineering |
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facultyofscienceandengineering |
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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 |
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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 |
2019-06-16T04:01:09Z |
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1763753139559727104 |
score |
11.021648 |