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 ${\mathcal{E}}_{\min }^{\alpha }(\mathbb{C})$, the Fréchet 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 $f\in {\mathcal{E}}_{\min }^{\alpha }(\mathbb{C})$ admits a unique decomposition $f(z)={\sum }_{n=0}^{\infty }{c}_n{s}_n(z)$, and the series converges in the topology of ${\mathcal{E}}_{\min }^{\alpha }(\mathbb{C})$. Within the context of a complex nuclear space $\Phi$ and its dual space ${\Phi }^{\prime }$, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on ${\Phi }^{\prime }$. In particular, for $\Phi ={\Phi }^{\prime }={\mathbb{C}}^n$ with $n\ge 2$, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space ${\Phi }^{\prime }$, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of ${\mathcal{E}}_{\min }^{\alpha }({\Phi }^{\prime })$ when $\alpha >1$. The latter result is new even in the one-dimensional case.

References in corpus (2)

• On a spectral representation for correlation measures in configuration space analysis
• An infinite dimensional umbral calculus

Cited by in corpus (2)

• Segal-Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
• Finite difference calculus in the continuum