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# Questions tagged [fourier-analysis]

The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

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### How to choose some $h$ so its Fourier transform supported in some set?

Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$ Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
1 vote
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### A parametrix construction for heat boundary value problem using Fourier transformation

Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3$ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
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### Sobolev estimates on domain with boundary

Could someone point me to a reference for the proof of the following Sobolev estimate $$\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})$$ for ...
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### Can the best constants in harmonic analysis be approximated in principle?

Consider the trivial example of Holder's inequality $\|f\|_p\,\|g\|_q\geq |fg|_1$ if $\frac{1}{p}+\frac{1}{q}=1, p,q\geq 1$ and $f,g$ are functions on $\mathbb{R}^n$. Let's suppose we don't know how ...
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### $\|\hat{f}\|_{L^q}< \infty \implies \left\| \|\chi_{n+(-1/2, 1/2]} \widehat{f}\|_{L^p_{\xi}} \right\|_{\ell^q_n}<\infty$

Suppose that support of $f:\mathbb R \to \mathbb R$ is compact set $K\subset \mathbb R.$ Assume that $\int_{\mathbb R} |\widehat{f}|^q d\xi <\infty.$ ($\widehat{\cdot}$ denote the Fourier ...
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### Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive ...
1 vote
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### Littlewood-Paley characterisation of Hölder regularity

I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...
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### Is this formula for 2D Fourier integral of diffraction kernel correct?

Well I have a function parametrized by $z$ $$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$ where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
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### Existence of a specific family of functions on an abelian group with vanishing properties on rank 2 subgroups

Fix a prime $p$, and let $W_0\subset W$ be an inclusion of a codimension one $\mathbb{F}_p$ vector spaces. Let $W_e$ denote a fixed nontrivial coset of $W_0$ in $W$. The question is whether there ...
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### Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?

Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let $$F_a(t) = \sum_{k \in \mathbb Z} f(t+ak)$$ be the ...
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### Real-analytic analogue of Schwartz functions

Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic ...
1 vote
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### Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis?
Nirenberg's paper On elliptic PDEs claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality  \...
Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f :\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold? 1- $-n\lneqq f_{\min}$ (where \$f_{\...