Gelfand-Shilov spaces for extended Gevrey regularity

TL;DR

This work develops Gelfand-Shilov spaces adapted to extended Gevrey regularity by employing a weight-matrix framework based on $M_p^{\tau,\sigma}=p^{\tau p^{\sigma}}$ with $\sigma>1$. It proves nuclearity of these spaces, establishes Fourier transform invariance, and provides symmetric characterizations, all within Roumieu/Beurling variants and $L^2$-based formulations. The paper also analyzes time-frequency representations (Grossmann-Royer, STFT, Wigner, and cross-ambiguity) acting on these spaces, proving closure and transform-invariance properties and connecting time-frequency analysis with extended Gevrey ultradifferentiability. Overall, it extends classical Gelfand-Shilov theory to broader global ultradifferentiable classes and demonstrates stability under ultradifferential operators and standard transforms, aided by Lambert-W-based growth controls and the weight-matrix framework.

Abstract

We consider spaces of smooth functions obtained by relaxing Gevrey-type regularity and decay conditions. It is shown that these classes fit well within the general framework of the weighted matrices approach to ultradifferentiable functions. We examine equivalent ways of introducing Gelfand-Shilov spaces related to the extended Gevrey regularity and derive their nuclearity. In addition to the Fourier transform invariance property, we present their corresponding symmetric characterizations. Finally, we consider some time-frequency representations of the introduced classes of ultradifferentiable functions.

Theorems & Definitions (27)

• Definition 2.1
• Lemma 2.2
• Remark 2.3
• Definition 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Remark 3.4
• Theorem 3.5