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0 votes

General procedure for inverse of an integral transform

I believe OP is asking: given a function $\varphi(\xi)$, can one find a function $f$ such that $\varphi(\xi) = \int_{a}^{b}f(x)g(x,\xi)dx$? This is essentially the theory of integral equations. Tables ...
6 votes

How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?

As Dan Romik points out, this technique is relatively old folklore by now. Terry Tao calls this the "tensor power trick" in a blog post dedicated to the subject; the two elementary ...
6 votes

How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?

An instance of this idea of killing an unwanted factor in an inequality by considering an inequality for $k$-th powers and then taking the limit as $k\to\infty$ appears in the proof of the Kraft-...
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3 votes

Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?

One can prove that under your assumptions $$A:=\int_{-\infty}^\infty|x|f(x)dx\geq 9/(4\pi),$$ but estimate is not exact. The proof is based on the formula $$A=-\lim_{y\to 0+}\frac{1}{\pi}\frac{d}{dy}\...
5 votes

Approximating a function by a convolution of given function?

This is an extended comment. Passing to Fourier transforms, we have the following problem: $\hat{f}$ and $\hat{g}$ are two functions analytic in some strip $-a<\mathrm{Im}\, z<a$, and bounded ...
5 votes

Fastest decay of Fourier transform of function of (one-sided or two-sided) exponential decay

The simplest result can be stated when both your conditions hold. If they are satisfied, your Fourier transform is analytic and bounded from above in the strip $-a_2<\mathrm{Im}\, s<a_1$, and ...

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