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

Abstract

We study the connection between STFT multipliers A1⊗mg1,g2 having windows g1,g2, symbols a(x,ω)=(1⊗m)(x,ω)=m(ω), (x,ω)∈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=A1⊗mg1,g2. 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 A1⊗mg1,g2, 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 A1⊗mg1,g2. As a by-product we prove necessary conditions for the continuity of anti-Wick operators A1⊗mg,g:Lp→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.