Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | , , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10400.2/8550 |
Resumo: | 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 Fréchet topological space of entire functions of exponential order $\alpha$ and minimal type. 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. |
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Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysisInfinite dimensional holomorphyNuclear and co-nuclear spacesPolynomials sequence of binomial typeSheffer operatorSheffer sequenceSpaces of entire functionsFor 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 Fréchet topological space of entire functions of exponential order $\alpha$ and minimal type. 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.ElsevierRepositório AbertoFinkelshtein, Dmitri L.Kondratiev, Yuri G.Lytvynov, EugeneOliveira, Maria JoãoStreit, Ludwig2019-09-30T11:10:35Z2019-11-012019-11-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.2/8550engFinkelshtein, Dmitri [et al.] - Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis. "Journal of Mathematical Analysis and Applications" [Em linha]. ISSN 0022-247X. Vol. 479, nº 1 (2019), p. 162-1840022-247X10.1016/j.jmaa.2019.06.021info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-16T15:30:24Zoai:repositorioaberto.uab.pt:10400.2/8550Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T22:48:34.540407Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
title |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis |
spellingShingle |
Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis Finkelshtein, Dmitri L. Infinite dimensional holomorphy Nuclear and co-nuclear spaces Polynomials sequence of binomial type Sheffer operator Sheffer sequence Spaces of entire functions |
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 |
Finkelshtein, Dmitri L. |
author_facet |
Finkelshtein, Dmitri L. Kondratiev, Yuri G. Lytvynov, Eugene Oliveira, Maria João Streit, Ludwig |
author_role |
author |
author2 |
Kondratiev, Yuri G. Lytvynov, Eugene Oliveira, Maria João Streit, Ludwig |
author2_role |
author author author author |
dc.contributor.none.fl_str_mv |
Repositório Aberto |
dc.contributor.author.fl_str_mv |
Finkelshtein, Dmitri L. Kondratiev, Yuri G. Lytvynov, Eugene Oliveira, Maria João Streit, Ludwig |
dc.subject.por.fl_str_mv |
Infinite dimensional holomorphy Nuclear and co-nuclear spaces Polynomials sequence of binomial type Sheffer operator Sheffer sequence Spaces of entire functions |
topic |
Infinite dimensional holomorphy Nuclear and co-nuclear spaces Polynomials sequence of binomial type Sheffer operator Sheffer sequence Spaces of entire functions |
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 Fréchet topological space of entire functions of exponential order $\alpha$ and minimal type. 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. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-09-30T11:10:35Z 2019-11-01 2019-11-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10400.2/8550 |
url |
http://hdl.handle.net/10400.2/8550 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Finkelshtein, Dmitri [et al.] - Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis. "Journal of Mathematical Analysis and Applications" [Em linha]. ISSN 0022-247X. Vol. 479, nº 1 (2019), p. 162-184 0022-247X 10.1016/j.jmaa.2019.06.021 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier |
publisher.none.fl_str_mv |
Elsevier |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
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1799135066335477760 |