Comparisons between Fourier and STFT multipliers: The smoothing effect of the short-time Fourier transform

We study the connection between STFT multipliers Ag1,g2 1 & OTIMES;m having windows g1, g2, symbols a(x, w) = (1 & OTIMES; m)(x, w) = m(w), (x, w) & ISIN; R2d, and the Fourier multipliers Tm2 with symbol m2 on Rd. We find sufficient and necessary conditions on symbols m, m2 and windows g1, g2 for the equality Tm2 = Ag1,g2 1 & OTIMES;m . For m = m2 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 Ag1,g2 1 & OTIMES;m, 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 Ag1,g2 1 & OTIMES;m. As a by-product we prove necessary conditions for the continuity of anti -Wick operators Ag,g 1 & OTIMES;m : Lp & RARR; Lq having multiplier m in weak Lr spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multipliers are better known as linear time invariant (LTI) filters. & COPY; 2023 Elsevier Inc. All rights reserved.