No Cover Image

Journal article 949 views 160 downloads

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

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

Journal of Mathematical Analysis and Applications, Volume: 479, Issue: 1, Pages: 162 - 184

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

  • Finalversionv4.pdf

    PDF | Accepted Manuscript

    Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND).

    Download (382.91KB)

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...

Full description

Published in: Journal of Mathematical Analysis and Applications
ISSN: 0022-247X
Published: Elsevier BV 2019
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa50766
Tags: Add Tag
No Tags, Be the first to tag this record!
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 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.
Keywords: Infinite dimensional holomorphy; Nuclear and co-nuclear spaces; Sequence of polynomials of binomial type; Sheffer operator; Sheffer sequence; Spaces of entire functions
Issue: 1
Start Page: 162
End Page: 184