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

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 Fourier Tm2 with symbol m2 on Rd. find sufficient necessary conditions m,m2 g1,g2 for equality Tm2=A1⊗mg1,g2. For m=m2 former holds only particular choices of window functions in modulation spaces, whereas it never occurs realm Lebesgue spaces. In general, multiplier A1⊗mg1,g2, also called localization operator, presents a smoothing effect due to so-called two-window short-time transform which enters definition A1⊗mg1,g2. As by-product we prove continuity anti-Wick operators A1⊗mg,g:Lp→Lq m weak Lr Finally, exhibit related results their discrete counterpart: this setting are Gabor better known as linear time invariant (LTI) filters.