# 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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### Meaning of Big O notation with Index $[O_{\epsilon}(n^{\epsilon})]$

I found this notation in a book and can't figure what it means. I haven't found any other examples. This is $$O_{\epsilon}(n^{\epsilon})$$ One can see it in context at page $186$ of book. Anyone know ...
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### When does Inverse Fourier transform look close to a positive definite function?

Let $G$ be a commutative locally compact group, and $\hat{G}$ be its dual group, consisting of all continuous characters (continuous homomorphisms from $G$ to the circle group $\mathbb{T}$) . I can ...
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### Is the function $(1+|\ln |x||)^{-1/2}$ on $\mathbb{R}$ a Fourier multiplier?

How to judge whether the function $(1+|\ln |x||)^{-1/2}$ is a Fourier multiplier on $\mathbb{R}$? Should I use some multiplier theorems?
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### Dual of $L \log L(\mathbb{R})$

Consider the space$$L\log L(\mathbb{R})=\left \{f\in L^1(\mathbb{R}):\int \limits _\mathbb{R}|f(x)|\log ^+|f(x)|\,dx<\infty \right \}.$$Is it known what its dual and predual spaces are? Also any ...
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### problem on $L^2$ (pointwise) convergence and Carleson's Theorem

I am having some problem on my arguments and I want to see where it fails. It deals with pointwise convergence in $L^2[-\pi,\pi]$: pointwise convergence is given by Carleson's Theorem (so it is a hard ...
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### Spherical Mean of Schwartz Function Decreases Rapidly - Stein, Shakarchi, Fourier Analysis.

I'm studying Stein and Shakarchi's Fourier Analysis book, and I'm stumped on page. 189, where it talks about the spherical mean of a function. The book defines the spherical mean of a complex-valued ...
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### Multi-dimensional Dirichlet-Dini criterion for Fourier series

Let $\mathbb I^d$ be the $d$-dimensional unit cube, and $f\in L^1(\mathbb I^d)$. Further let $x\in\mathbb I^d$ and assume that (some representative of) $f$ is differentiable at $x=(x_1,\dotsc, x_d)$ (...
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### Convolution inequality $\|f\star g\|_p \le \|f\|_1 \|g\|_p$ on locally compact group

Consider the following fragment from Folland's book "A course in abstract harmonic analysis". Here $G$ is a locally compact Hausdorff group and the $L^p$-spaces are considered w.r.t. a left ...
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### Determine where function is discontinuous by its fourier coefficients

The fourier coefficients of ${f} \in C_{PW}^0$ are given by $\widehat f(n)=(-i)^n(\frac{3i}{4n}+\frac{1}{2\pi n^2})$ for $n \ne 0$. Find where $f$ is discontinuous and what is its jump there? it's ...
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### A question on Banach space involving an invariant mean on $\mathbb Z$

Let $E$ be a Banach space and suppose $T\in Aut(E)$ satisfies the condition $\|T^n\|\leq C$ for all $n\in \mathbb Z$ for some fixed $C\in \mathbb R$. I have to show that there is an equivalent norm on ...
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### On a Stopping Time Argument in Harmonic Analysis

Quite often there is a common trick in harmonic analysis like the following: Assume that $0<S(x)<\infty$ for all $x\in\mathbb{R}^{n}$. Consider the set \begin{align*} \Omega_{k}&=\{x\in\...
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### Integrating on a subspace of a scalar field = integrating on the orthogonal subspace of the Fourier transform

Suppose you have some function $f: \Bbb R^n \to \Bbb R$. You also have an injective linear transformation $A$ mapping from $\Bbb R^m \to \Bbb R^n$, with $m < n$, so that the image of $A$ is some $m$...
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### Fourier transforms of multidimensional "delta subspace" distributions

Sometimes, when working in higher-dimensional vector spaces, one runs into higher-dimensional delta distributions whose support is a linear subspace. A fairly common example from $\Bbb R^2$ would be, ...
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### Pontryagin Dual and Continuity involving w* topology

For $G$, a locally compact group, we've defined $\hat{G}$ to be the group of all continuous homomorphisms from $G$ to the torus $\mathbb{T}$. With this definition, $\hat{G}\subseteq L^\infty(G)$ as ...
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### Uniformity of a positive measure

Let $\lambda(z)$ be a nonnegative function in the punctured unit disc $B_1(0)\backslash \{0\}$, if there exists a constant $C>0$ such that for any disc $B_r(p)\subset B_{1}(0)\backslash \{0\}$, we ...
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### How to prove a lemma about $\lim _{\lambda \rightarrow \infty} e^{-\lambda t} \sum_{k \leq \lambda x} \frac{(\lambda t)^{k}}{k !}$? [duplicate]
I recently came across the following lemma while learning harmonic analysis, but don't know how to prove it by using analytical methods. Lemma: For all $t \geq 0, x>0$, we have that \begin{...