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

TL;DR

This paper characterizes certain Gelfand-Shilov subspaces and Pilipovi{\'c} spaces via the Bargmann transform, using estimates of harmonic oscillator powers to describe their structure and images.

Contribution

It provides new characterizations of Gelfand-Shilov and Pilipovi{\'c} spaces through the Bargmann transform and harmonic oscillator estimates, extending understanding of their structure.

Findings

Characterization of $\\maclJ(\rr d)$ and $\\maclJ_0(\rr d)$ via harmonic oscillator estimates.

Description of images of Pilipovi{\'c} spaces under the Bargmann transform.

Extension of Gelfand-Shilov space analysis using harmonic oscillator bounds.

Abstract

We consider the counter images $\maclJ (\rr d)$ and $\maclJ _0(\rr d)$ 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 Pilipovi{\'c} spaces $\bsycalS _s(\rr d)$ and $\bsySig _s(\rr d)$ when $0<s<1/2$ and deduce their images under the Bargmann transform.