Modulation spaces, multipliers associated with the special affine Fourier transform

TL;DR

This paper explores the properties of the special affine Fourier transform (SAFT), introducing a new modulation space and establishing key theorems like the H"{o}rmander multiplier and Littlewood-Paley theorems in this context.

Contribution

It introduces a new modulation space linked to SAFT and proves fundamental theorems connecting operators, multipliers, and analysis in this setting.

Findings

Established the connection between bounded linear operators and A-convolution operators.

Proved the H"{o}rmander multiplier theorem for SAFT.

Proved the Littlewood-Paley theorem for SAFT.

Abstract

We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space $\boldsymbol {M}^{r,s}_A$ in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with $A$-translation, then it is a $A$-convolution operator. We also establish H\"{o}rmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.