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Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis / Dmitri Finkelshtein, 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:

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

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...

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Published in: Journal of Mathematical Analysis and Applications 0022-247X Elsevier BV 2019 https://cronfa.swan.ac.uk/Record/cronfa50766 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. Infinite dimensional holomorphy; Nuclear and co-nuclear spaces; Sequence of polynomials of binomial type; Sheffer operator; Sheffer sequence; Spaces of entire functions 1 162 184