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

We study the connection between STFT multipliers $A^{\f_1,\f_2}_{1\otimes m}$ having windows $\f_1,\f_2$, symbols $a\phas=(1\otimes m)\phas=m(\o)$, $\phas\in\rdd$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on $\rd$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $\f_1,\f_2$ for the equality $T_{m_2}= A^{\f_1,\f_2}_{1\otimes m}$. 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 $A^{\f_1,\f_2}_{1\otimes m}$, also called localization operator, presents a smoothing effect due to the so-called \emph{two-window short-time Fourier transform} which enters in the definition of $A^{\f_1,\f_2}_{1\otimes m}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A^{\f,\f}_{1\otimes m}: L^p\to L^q$ 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.