Paley-Wiener properties for spaces of entire functions

Paley–Wiener properties for spaces of power series expansions

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We extend Paley-Wiener results in the Bargmann setting deduced in [8] to larger class of power series expansions. At the same time we deduce characterisations of all Pilipović spaces and their distributions (and not only of low orders as in [8]).

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“…when χ above is fixed. In fact, in [8], similar characterizations are deduced for all A ♭σ (C d ) and A 0,♭σ (C d ) with σ ≤ 1 as well as for the set A s (C d ) when s ∈ [0, 1 2 ) which consists of all F ∈ A(C d ) such that |F (z)| e r(log z ) 1 1−2s . For example, let χ be the characteristic function of a polydisc, centered at origin in C d .…”

“…In [8,13] a type of Paley-Wiener results were deduced for certain spaces of entire functions, where the usual Fourier transform were replaced by the reproducing kernel Π A of the Bargmann transform. (See [6] and Section 1 for notations.)…”

“…for some characteristic function χ of a polydisc D, centered at origin, and a function F which is defined and analytic in a neighbourhood of D. The spaces A 0,♭ 1 (C d ) is also characterized in [8] by…”

“…For example, let χ be the characteristic function of a polydisc, centered at origin in C d . Then it is proved in [8] that…”