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Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis
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Advisor(s)
Abstract(s)
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.
Description
Keywords
Infinite dimensional holomorphy Nuclear and co-nuclear spaces Polynomials sequence of binomial type Sheffer operator Sheffer sequence Spaces of entire functions
Pedagogical Context
Citation
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
Publisher
Elsevier