# Questions tagged [harmonic-analysis]

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

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### What are some topics in representation theory of locally compact groups?

I am currently studying representations of locally compact groups and I find it a really interesting subject so I would like to know more. First I would like to ask if this is an active area of ...
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### Probabilistic Hausdorff-Young Type Inequality

Let $1 \leq p <2$ and let $q$ be the Holder conjugate of $p$ so that $\frac{1}{p} + \frac{1}{q} = 1$. Show that for any $\epsilon >0$, there exists a Schwartz function $f \in S(\mathbb{R}^d)$, ...
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### BMO space and representation theory

Does the BMO function have any applications in representation theory? Thanks!
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### What is the Schwartz-type space for Mellin transform?

It is well known that for $f\in S(\mathbb R)$, the Schwartz space, one can assert that $f^{(a)}$, $Ff^{(a)}$ (the Fourier transform of $f^{(a)}$) are also in $S(\mathbb R)$ for any $a=0,1,2,\ldots$. ...
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### Is a homogeneous Banach space on $\mathbb T$ always well defined?

A homogeneous Banach space $B$ on group $\mathbb T=\mathbb R/2\pi\mathbb Z$ is a linear subspace $B$ of $L^1(\mathbb T)$ having a norm $\|\ \|_B\ge\|\ \|_{L^1}$ under which it is a Banach space, and ...
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### Is$\prod_{n=1}^{\infty} (x-n\pi)$ a factor of $\sin x$?

As it holds for a polynomial it should be true for a power series too. Why not?
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I am interested in understanding some general properties of complete invariant maps. An invariant is a map $$f: X \rightarrow Y$$ defined over objects $X$ and a transformation $\tau$ such that $f(... 1answer 29 views ### dimension of a vector space of harmonic functions among polynomials in two variables I'm searching for harmonic functions inside a set of HOMOGENIOUS polynomials in two variables. Let's say that $$P_n = \{\sum_{i+j = n} a_{ij} x^i y^j \quad|\quad a_{ij} \in \mathbb{R}\}$$ Let's write$...
Suppose $G$ is a discrete abelian group. Show that $\hat{G}$ is compact. This exercise is the converse of this question: The Pontryagin dual of a compact abelian group is discrete An example of this ...