The Grossmann-Royer transform, Gelfand-Shilov spaces, and continuity properties of localization operators on modulation spaces

TL;DR

This paper reviews localization operators on modulation spaces using the Grossmann-Royer transform, providing new insights, simplified proofs, and detailed analysis of Gelfand-Shilov spaces and their duals in phase space.

Contribution

It introduces a new approach based on the Grossmann-Royer transform, simplifying proofs and extending properties to Gelfand-Shilov spaces within time-frequency analysis.

Findings

Grossmann-Royer transform simplifies proofs of known results.

Gelfand-Shilov spaces are limits of modulation spaces.

Localization operators defined via Grossmann-Royer match traditional definitions.

Abstract

This paper offers a review of the results concerning localization operators on modulation spaces, and related topics. However, our approach, based on the Grossmann-Royer transform, gives a new insight and (slightly) different proofs. We define the Grossmann-Royer transform as interpretation of the Grossmann-Royer operator in the weak sense. Although such transform is essentially the same as the cross-Wigner distribution, the proofs of several known results are simplified when it is used instead of other time-frequency representations. Due to the importance of their role in applications when dealing with ultrafast decay properties in phase space, we give a detailed account on the Gelfand-Shilov spaces and their dual spaces, and extend the Grossmann-Royer transform and its properties in such context. Another family of spaces, modulation spaces, are recognized as appropriate background for time-frequency analysis. In particular, the Gelfand-Shilov spaces are revealed as projective and inductive limits of modulation spaces. For the continuity and compactness properties of localization operators we employ the norms in modulation spaces. We define localization operators in terms of the Grossmann-Royer transform, and show that such definition coincides with the usual definition based on the short-time Fourier transform.