# Questions tagged [harmonic-analysis]

Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.

1,178 questions
Filter by
Sorted by
Tagged with
108 views

### Urysohn's lemma for Bochner functions?

Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used: If $U$ is an open ...
203 views

1 vote
33 views

### Stability of Hajłasz-Sobolev class under post-composition

Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev? Assumptions/Setup Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
56 views

### Extracting the point mass measure of some type of positive measures

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals. Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
62 views

### An equation in the convolution measure algebra on reals

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals. Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
94 views

344 views

### Unifying two definitions of $L^\infty$

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$. Definitions: A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
1 vote
77 views

### Inequality on the dual space of $H^s$

Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ? For instance, assume ...
193 views

### Probability measure on the boolean cube with small support and small Fourier transform

[Edit: added details on the Reed-Muller codes] Are there explicit (non random) constructions of probability measures on $D_N = \{0,1\}^N$ with support of size $O(N)$ and with all nontrivial Fourier ...
49 views

Let $(X,\mathcal{B}(X),(\mathcal{A}_n)_{n=1}^N,\mu)$ be a filtered probability space with $\mathcal{A}_N=\mathcal{B}(X)$ and let $H$ be the space of $\mathcal{A}_{\cdot}$-adapted martingales $m_{\cdot}... 3 votes 0 answers 107 views ### Erdős–Turán inequality for complex numbers Consider the following set of complex numbers in the upper half plane: $$\{ic_n \pm \gamma_n: 0 \leq n \leq N, \hspace{1 mm} c_0=\gamma_0=0, c_n,\hspace{1 mm} \gamma_n>0\}.$$ Assume that this set ... 3 votes 1 answer 128 views ### Fourier multipliers and transference on cyclic groups It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on$L^p(\mathbb{R}^n)$then by transference the similar operator is also bounded on$L^p(\...
A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions. This MSE question asked ...