Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

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 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.