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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
