Questions tagged [harmonic-analysis]

Question 1

The question at hand is the following: Let $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be real–valued such that $\psi \ge 0$ and $\int_{\mathbb{R}^d} \psi(x)\,dx = 1$. Moreover, let $f \in L^1(\...

Question 2

If $T$ is a $1\times \delta$ rectangle in $\mathbb{R}^2$ for a $\delta>0$, where $\delta$ is small, then you one take a Schwartz function $f$ so that $f\geq 1$ on $T$, and $\hat{f}$ is compactly ...

Question 3

This year I have been studying the paper "The proof of $l^{2}$ decoupling conjecture" by Bourgain and Demeter. by Bourgain and Demeter. I have been able to understand every part of the proof ...

Question 4

I came across this problem when studying spectral multipliers concerning the Laplace operator on Euclidean spaces. Denote $L = - \Delta$ as the Laplace operator on $\mathbb{R}^n$. Given $F \in L^2_s(\...

Question 5

My problem is the following: Firstly, let $G$ be a locally compact abelian group and denote by $L^{1}(G)$ the Banach $*$-algebra with respect to the convolution of functions as the multiplication and ...

Question 6

I'm trying to do a Grafakos exercise that asks me to show that the space $BMO(\mathbb{R}^n)$ is complete. To do this, it suggests using the following approach: Let $f\in BMO(\mathbb{R}^n)$. For any $\...

Question 7

The following estimate arises in the proof of Tomas-Stein restriction theorem. $$ \sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1} $$ The estimate is very intuitive, and I have a ...

Question 8

I am searching for a good reference for finding the fourier inversion theorem for compact (abelian, or non abelian) groups (we shall denote such a group by $G$). In particular, I would like to see a ...

Question 9

I want to prove that $C^{*}(G)\cong \oplus_{\pi\in\hat{G}}M_{dim\pi}(\mathbb{C})$. Of course, one wants to use the Peter-Weyl theorem for this (the version, which states that $L^{2}(G)\cong \oplus_{\...

Question 10

Let $\sigma_I=\{e^{2i\pi t}:t\in I\}$ and $\sigma_J=\{e^{2\pi i t}:t\in J\}$ be two disjoint subarcs on the the first quadrant of unit circle of arc length $\theta$, where $I,J\subseteq [0,1/4]$ of ...

Question 11

I’m trying to understand a step in Appendix A.1 of Bejenaru and Herr, The cubic Dirac equation: small initial data in $H^1(\mathbb{R}^3)$. The paper proves global well-posedness for the cubic Dirac ...

Question 12

I'm struggling to provide a proper conceptual reason for what is going on here. For background, I was taught in school the three major transforms, Laplace first, Fourier second and Mellin last (but ...

Question 13

What is the Haar measure on the Bohr compactification $b\mathbb{Z}$ of the integers? We (my collaborators and I) suspect that it is $$ \int_{b\mathbb{Z}} d\mu_H f(n) = \lim_{N\to\infty}\frac{1}{2N+1} \...

Question 14

Consider the Hilbert space $\mathcal{H}$ = $L^2(\mathbb{A}_{\mathbb{Q}}^{\times} / \mathbb{Q}^{\times})$, where $\mathbb{A}_{\mathbb{Q}}^{\times}$ is the idèle group of $\mathbb{Q}$ equipped with its ...

Question 15

I want to show that $C^{*}(G)\cong C_{0}(\hat{G})$, where G is a locally compact abelian group. I alredy know that, since G is abelian $C^{*}(G)$ is also abelian and that one applies the Gelfand ...

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Questions tagged [harmonic-analysis]

Approximation by convolution with Schwartz function

Fourier transform of indicator of curved tube

Ball inflation inequality on restriction theory, l^{2} decoupling

L1 boundedness of some spectral multiplier

Why are $\widehat{C^{*}(G)}$ and $\widehat{L^{1}(G)}$ homeomorphic?

How to prove that Cauchy in BMO then is Cauchy in $L^1$

Area estimate of the intersection of ball and sphere

Fourier inversion theorem for compact groups

Classification of $C^{*}(G)$ for a compact group $G$

On the convolution identity of a sub arc of circle and the open set which is thickened epsilon amount of another subarc in circle.

Can Young's inequality give a pointwise bound for a convolution?

Why is the Reciprocal Log Transform so "un-creative"? Why does it seem to "interpolate" between Fourier, Mellin and Laplace?

What is the Haar measure on the Bohr compactification $b\mathbb{Z}$ of the integers?

What do self-adjoint operators on $L^2(\mathbb{A}_{\mathbb{Q}}^{\times} / \mathbb{Q}^{\times})$ look like?

Classification of an LCA group C*-algebra

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