The Bargmann transform and powers of harmonic oscillator on Gelfand–Shilov subspaces

Abstract

Abstract. We consider the counter images J (R d ) and J 0 (R d ) of entire functions with exponential and almost exponential bounds, respectively, under the Bargmann transform, and we characterize them by estimates of powers of the harmonic oscillator. We also consider the Pilipović spaces S s (R d ) and Σ s (R d ) when 0 < s < 1/2 and deduce their images under the Bargmann transform. IntroductionThe aim of the paper is to characterize the images of the Pilipović spaces Σ s (R d ) and S s (R d ) under the Bargmann transform when s < 1/2, as well as the test function spaces, in terms of estimates of powers of the harmonic oscillator. The set J (R d ) consists of all f ∈ S (R d ) such that their Hermite series expansions are given bywhere

Tensor products for Gelfand–Shilov and Pilipović distribution spaces

“…Also let A 0,1/2 (C d ) be the set of all F ∈ A(C d ) such that |F (z)| e r|z| 2 for all r > 0. Then it is proved in [5,14]…”

Paley–Wiener properties for spaces of entire functions

“…which shall be reached by modifying the proof of (15) in [4]. We have ϑ r (α) 2 ∞ e e (r 1 −r)(log t) θ g r 1 ,α (t) dt sup t≥e (g r 1 ,α (t)) ∞ e e (r 1 −r)(log t) θ dt ≍ sup t≥e (g r 1 ,α (t)).…”

“…A r,s (C d ), The spaces in (0.1) appear naturally when considering the Bargmann transform images of an extended class of Fourier invariant Gelfand-Shilov spaces, called Pilipović spaces (see [4,12]).…”

Paley–Wiener properties for spaces of power series expansions

“…for some h, r > 0 (for every h, r > 0) (cf. [4,13] and Section 1). We remark that for s ≥ 1 2 , A s (C d ) and A 0,s (C d ), defined in this way, are still spaces of entire functions, but with other types of estimates, compared to the case s < 1 2 considered above (cf.…”