Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

Peter Balazs; Federico Bastianoni; Elena Cordero; Hans G. Feichtinger; Nina Schweighofer

doi:10.1016/j.jmaa.2023.127579

paper

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

arXiv:2203.01142 · doi:10.1016/j.jmaa.2023.127579

Abstract

We study the connection between STFT multipliers $A_{1 \otimes m}^{g_{1}, g_{2}}$ having windows $g_{1}, g_{2}$ , symbols $a (x, ω) = (1 \otimes m) (x, ω) = m (ω)$ , $(x, ω) \in R^{2 d}$ , and the Fourier multipliers $T_{m_{2}}$ with symbol $m_{2}$ on $R^{d}$ . We find sufficient and necessary conditions on symbols $m, m_{2}$ and windows $g_{1}, g_{2}$ for the equality $T_{m_{2}} = A_{1 \otimes m}^{g_{1}, g_{2}}$ . For $m = m_{2}$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A_{1 \otimes m}^{g_{1}, g_{2}}$ , also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of $A_{1 \otimes m}^{g_{1}, g_{2}}$ . As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A_{1 \otimes m}^{g, g} : L^{p} \to L^{q}$ having multiplier $m$ in weak $L^{r}$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.

39 pages, 3 figures