PUB - Publikationen an der Universität Bielefeld

For certain Sheffer sequences (s(n))(n=0)(infinity) on C, Grabiner (1988) proved that, for each ( )alpha is an element of [0, 1), the corresponding Sheffer operator z(n) bar right arrow s(n)(z) extends to a linear self-homeomorphism of epsilon(alpha)(min)(C), the Frechet 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 is an element of epsilon(alpha)(min) (C) admits a unique decomposition f (z) =Sigma(infinity)(n=0) CnSn(z), and the series converges in the topology of epsilon(alpha)(min) (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' = C-n with n >= 2, 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 epsilon(alpha)(min) (Phi') when alpha > 1. The latter result is new even in the one-dimensional case. (C) 2019 Elsevier Inc. All rights reserved.