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

Abstract

We study the connection between STFT multipliers A g1,g2 1⊗m having windows g 1 , g 2 , symbols a(x, ω) = (1 ⊗ m)(x, ω) = m(ω), (x, ω) ∈ R 2d , and the Fourier multipliers T m2 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 equality1⊗m . 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 g1,g2 1⊗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 A g1,g21⊗m . As a by-product we prove necessary conditions for the continuity of anti-Wick operators A g,g 1⊗m : L p → 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.