Fix $f \in L^2(\mathbb{R})$ s.t. $||f||_2=1.$ When will $$V_f f (x, \omega)=\int_{\mathbb{R}} f(t)\overline{f(t-x)}e^{-2 \pi i t\omega}dt$$ the STFT of $f$ with respect to the window $f$ be in $L^1(\mathbb{R}^2)?$ Can we get a bound on the $L^1$ norm in this case?
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