Journal article 882 views 156 downloads
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis
Dmitri Finkelshtein 
,
Yuri Kondratiev,
Eugene Lytvynov ,
Maria João Oliveira,
Ludwig Streit
,
Yuri Kondratiev,
Eugene Lytvynov ,
Maria João Oliveira,
Ludwig Streit
Journal of Mathematical Analysis and Applications, Volume: 479, Issue: 1, Pages: 162 - 184
Swansea University Authors:
Dmitri Finkelshtein , Eugene Lytvynov
-
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DOI (Published version): 10.1016/j.jmaa.2019.06.021
Abstract
For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $\alpha\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$, the Fr\'echet topological...
| Published in: | Journal of Mathematical Analysis and Applications |
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| ISSN: | 0022-247X |
| Published: |
Elsevier BV
2019
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa50766 |
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2020-07-20T15:46:50.6422210 v2 50766 2019-06-08 Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis 4dc251ebcd7a89a15b71c846cd0ddaaf 0000-0001-7136-9399 Dmitri Finkelshtein Dmitri Finkelshtein true false e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2019-06-08 SMA For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $\alpha\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$, the Fr\'echet topological space of entire functions of order at most $\alpha$ and minimal type (when the order is equal to $\alpha>0$). In particular, every function $f\in \mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$ admits a unique decomposition $f(z)=\sum_{n=0}^\infty c_n s_n(z)$, and the series converges in the topology of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$. Within the context of a complex nuclear space $\Phi$ and its dual space $\Phi'$, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on $\Phi'$. In particular, for $\Phi=\Phi'=\mathbb C^n$ with $n\ge2$, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space $\Phi'$, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\Phi')$ when $\alpha>1$. The latter result is new even in the one-dimensional case. Journal Article Journal of Mathematical Analysis and Applications 479 1 162 184 Elsevier BV 0022-247X Infinite dimensional holomorphy; Nuclear and co-nuclear spaces; Sequence of polynomials of binomial type; Sheffer operator; Sheffer sequence; Spaces of entire functions 1 11 2019 2019-11-01 10.1016/j.jmaa.2019.06.021 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-07-20T15:46:50.6422210 2019-06-08T10:46:36.3292803 Dmitri Finkelshtein 0000-0001-7136-9399 1 Yuri Kondratiev 2 Eugene Lytvynov 0000-0001-9685-7727 3 Maria João Oliveira 4 Ludwig Streit 5 0050766-08062019105307.pdf Finalversionv4.pdf 2019-06-08T10:53:07.5670000 Output 346890 application/pdf Accepted Manuscript true 2020-06-10T00:00:00.0000000 Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND). true eng |
| title |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
| spellingShingle |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis Dmitri Finkelshtein Eugene Lytvynov |
| title_short |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
| title_full |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
| title_fullStr |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
| title_full_unstemmed |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
| title_sort |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
| 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 Ludwig Streit |
| format |
Journal article |
| container_title |
Journal of Mathematical Analysis and Applications |
| container_volume |
479 |
| container_issue |
1 |
| container_start_page |
162 |
| publishDate |
2019 |
| institution |
Swansea University |
| issn |
0022-247X |
| doi_str_mv |
10.1016/j.jmaa.2019.06.021 |
| publisher |
Elsevier BV |
| document_store_str |
1 |
| active_str |
0 |
| description |
For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $\alpha\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$, the Fr\'echet topological space of entire functions of order at most $\alpha$ and minimal type (when the order is equal to $\alpha>0$). In particular, every function $f\in \mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$ admits a unique decomposition $f(z)=\sum_{n=0}^\infty c_n s_n(z)$, and the series converges in the topology of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$. Within the context of a complex nuclear space $\Phi$ and its dual space $\Phi'$, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on $\Phi'$. In particular, for $\Phi=\Phi'=\mathbb C^n$ with $n\ge2$, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space $\Phi'$, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\Phi')$ when $\alpha>1$. The latter result is new even in the one-dimensional case. |
| published_date |
2019-11-01T04:02:23Z |
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1763753216812515328 |
| score |
11.013686 |