All Questions

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...

Let $\phi:\mathbb{R}\to\mathbb{R}$ be an even function with support on $[-1,1]$. Assume that it is in $L^1\cap L^2$ and that its Fourier transform is also in $L^1\cap L^2$. Assume as well that $|\phi|...

This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish. A nice remark by Christian ...

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...

This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction? Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...