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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A sequence of harmonic functions on $\mathbb{R}^{2}$ converging in distribution must necessarily converge locally uniformly to an harmonic function.

I am self studying the Rudin's book of Functional Analysis and I stumbled upon this problem. Given a sequence $\{ f_{j} \}_{j}$ of harmonic functions on an open set $\Omega$ of $\mathbb{R}^{2},$ if $\{...
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The miraculous nature of the "matrix coefficients" $\langle Tv,v \rangle$ (especially in the context of positive type functions)

$\newcommand{\ak}[1]{\langle #1 \rangle}$I've noticed that for linear operators $T$ and an inner product $ \ak{\bullet, \bullet }$, the expression $\ak{ Tv,v}$ tends to show up a lot. For instance, it ...
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Absolute value of functions in $H^1(\mathbb R^n)$

Let $f$ be a function in the atomic Hardy space $H^1_{at}(\mathbb R^n)$. That is, there exists a sequence of atoms $a_j$ satisfying supp $a_j \subset B_j$ for some ball $B_j$, $\int a_j dx = 0$, $\...
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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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Boundedness of singular integral operator on normalized bump functions implies boundedness on Schwartz functions controlled by suitable seminorms

In the book "Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory integrals" by E. Stein, the author considers on page 294 a singular integral operator $T:\mathcal{S}(\...
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Is a certain map $\mathbb{R}\to \widehat{\mathbb{R}}$ a homeomorphism?

I'm studying harmonic analysis, and I'm trying to understand the fact that $$\phi: \mathbb{R}\to \widehat{\mathbb{R}}: s \mapsto (t \mapsto \exp(2\pi i st))$$ is an isomorphism of topological groups (...
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Question about a Corollary of Grafakos's "Modern Fourier Analysis"

The question is about the proof of Corollary 2.2.5 in Grafakos's book "Modern Fourier Analysis". And the statement can be writen as follows: Let $\phi,\psi \in \mathcal{S}(\mathbb{R}^n)$ and ...
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Prove a relation involving the Laplace–Beltrami operator and spherical harmonics

Let $P_ℓ$ denote the space of complex-valued homogeneous polynomials of degree $ℓ$ in $n$ real variables, here considered as functions ${\displaystyle \mathbb {R} ^{n}\to \mathbb {C} }$. Let $A_ℓ$ ...
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Prove the orthogonal decomposition of the space of spherical harmonics

Let $\mathbb{S}^{n}$ denote the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ endowed with the standard metric. For $k=0,1, \cdots$, denote by $\mathscr{H}_{k}^{n}$ the space of spherical ...
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Is there transfomation that transform the image to a space that I can use a small amount of function of fit it?

Now what I want to do is that I want to reprentation(just fitting) a image with neural network with as less error as possible,like paper NeRF(neural reprentation field) and FFN and Sire. The input is ...
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Completeness of weighted $L^p$ spaces

For $\infty >p>1$ consider the weighted $L^p$ spaces $(L^p(\mathbb{R}^n),\omega dx)$ where $\omega$ is some nonnegative weight. Is it true that $(L^p(\mathbb{R}^n),\omega dx)$ is complete iff $\...
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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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Lower Bound on Oscillatory Integral Related to Holder Functions

Let $\alpha\in(0,1]$ and $f\in C^{0,\alpha}([0,1])$ with $f(0)\neq0$ and $\text{supp } f\subset [0,1/2]$. What is the asymptotic behavior of the oscillatory integral as $\lambda\to+\infty$? $$I(\...
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Let $G$ be locally compact abelian, $T_2$ and $f\in L^1(G)$. Define $\mu(A)=\int_A f(x)\ dx$. Prove $\lVert \mu\rVert=\lVert f\rVert_1$

Here $\mu$ becomes a complex measure and $\lVert \mu\rVert =|\mu|(G)$ is the total variation norm of $\mu$. We have to show $|\mu|(G)=\int\limits_G |f(x)|\ dx$ Let $\{A_n\}$ be a partition of $G$. ...
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7 votes
2 answers
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Showing that the best approximating linear map for a Lipschitz function is also Lipschitz

For $d,n \in \mathbb{N}$ with $1 \leq d<n$, let $f: \mathbb{R}^d \to \mathbb{R}^n$ be a Lipschitz map with some constant $L \geq 1$. Let $B=B(0,r)$ for some $r>0$ and define the quantity $$ \...
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Induced representations: question about proof in Folland's book

Consider the following fragment from Folland's book "A course in abstract harmonic analysis": 6.1 The Inducing Construction Let $G$ be a locally compact group, $H$ a closed subgroup, $q : G ...
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Can a complex Radon measure be approximated by compactly supported Radon measures?

Let $G$ be an (abelian) locally compact Hausdorff group. Consider the following fragment from Folland's text "A course in abstract harmonic analysis" (second edition, p102). Why is the ...
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Why is $\phi_\mu(x)= \int_{\widehat{G}} \xi(x)d\mu(\xi)$ a continuous map?

Let $G$ be a locally compact Hausdorff abelian group. Let $\widehat{G}$ be its dual group, consisting of the unitary characters $G \to \mathbb{T}$. If $\mu \in M(\widehat{G})$ (= complex Radon ...
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How to solve this estimate in Grafakos' book?

Prove that for all $1<p<\infty $ there exist a constant $A_p>0$ such that for every $C^2_0(\mathbb{R}^2)$(twice continuously differentiable with compact support complex value function) such ...
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What function has $x \hat{f} (x)$ as its Fourier transform?

I have a function $f$ such that $$ f(y) = \int_{- \infty}^{\infty} \hat{f}(x)\,e^{2 \pi i x y} \,{\rm d}x. $$ I know the function $f$, but what I have is $$ G(y) = \int_{- \infty}^{\infty}x \hat{f}(...
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Understanding the principal value of the Beurling Transform

In doing research on Beltrami equations I am attempting to familiarize myself with the Beurling Transform, which is given as a principal value. Given $\phi \in C_0^\infty(\mathbb{C})$ we define the ...
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What are the best Books about abstract harmonic analysis? [closed]

I'm going to study abstract harmonic analysis and I need to some good reference about this. I'm familiar to topological group and haar measure. Can anyone help me?
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What does 'essentially supported' mean in uncertainty principle?

The uncertainty princple roughly says, in a heuristic way, that if a function $f$ is supported on a rectangle $T$, then its Fourier transform $\hat{f}$ is 'essentially supported' on the dual rectangle ...
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$H^2$ does not contain any rational functions with poles on the unit circle

The Hardy-hilbert space, $H^2$, consists of all analytic functions having power series representations with square-summable complex coefficients. That is, $$H^2=\{f : f(z)=\sum_{n=0}^{\infty} a_n z^ns....
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2 votes
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Rotationally invariant function after tensorizing

Suppose that $f$ is a real valued function such that $$f(x)f(y)=f((x+y)/\sqrt 2) f((x-y)/\sqrt 2)$$ for all real numbers $x,y$. Then is it necessarily true that $f(x)=Ae^{bx^2}$ for some real numbers $...
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4 votes
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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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2 votes
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Validity of Poisson summation formula

Let $\psi(x)$ be a smooth function with compact support. Let $f$ be a real valued function defined on $\mathbb{R}^n$. Then the Poisson summation formula holds $$ \sum_{x \in \mathbb{Z}^n} \psi(x) e^{2 ...
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$\int_{|y|>\epsilon}\frac{|\Omega(y/|y|)|}{|y|^n}|f(x-y)|\,dy<\infty \text{ for a.e.} x\in \mathbb R^n$, where $f\in L^p$ and $\Omega\in L^1(S^{n-1})$

Let $\Omega\in L^1(S^{n-1})$, where $S^{n-1}$ is the unit sphere in $\mathbb R^n$. Let $f\in L^p(\mathbb R^n)$ for some $p\in[1,\infty)$. For each $\epsilon>0$, prove that $$\int_{|y|>\epsilon}\...
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2 votes
3 answers
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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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1 answer
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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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3 votes
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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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Is this simple operator bounded from $L^p([1,2])$ to $L^p([0,1])$?

Let $f\in L^{p}([1,2])$ with $p>1$, and define $$Tf(x):= \int_{1<y<2}\frac{f(y)}{|x-y|}dy,\quad 0<x<1.$$ I am trying to prove/disporove that $$\|Tf\|_{L^p([0,1])}\leq C \|f\|_{L^p([1,2])...
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Spectral theory over 2D torus

I want to write a python simulation that requires the eigenfunctions and eigenvalues of the Laplace-Beltrami operator over the torus: $$ x=(R-rcos\theta)cos\phi , y=(R-rcos\theta)sin\phi , z=rsin\...
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1 vote
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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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2 votes
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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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1 vote
1 answer
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Mean value formula for harmonic functions on product of spheres

Is there a mean value formula for harmonic functions over product of spheres? I mean, let us consider $\mathbb{R}^4$ with coordinates $\alpha$, $\beta$, $\gamma$ and $\delta$ and the laplacian $\Delta:...
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Estimation of the modulus of the Fourier series derivative.

Assume$|\hat{f}(j)|\le K e^{-|j|^{\frac1\alpha}},\alpha>0,K>0.$Show that $f$ is infinitely differentiable and $$|f^{(n)}(t)|\le K_1 e^{an} n^{an}$$ for some constant $a$ and $K_1$. This question ...
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Dual of homogeneous Besov space

I am trying to prove a trace theorem from the homogeneous Besov space $\overset{\cdot}{B}_{2,1}^{1/2}(\mathbb{R}^n)$ to $L^2(K)$, where $K$ is a compact euclidean hypersurface. The homogeneous Besov ...
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1 vote
1 answer
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Let $G=SO(3)$ and $\theta:G\to G$ defined by $\theta(g)=JgJ$ where $J=\text{diag}(-1,1,1)$. Prove that, $\theta(g)\in Kg^{-1}K$ where $K=SO(2)$

Here $K=\left\{\begin{pmatrix}1 & 0\\ 0 & T_1\end{pmatrix}:\ T_1\in SO(2)\right\}$. $K$ is a compact subgroup of $G$. The above problem is part of the proof of $(G,K) $ is Gelfand pair. I've ...
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Cotlar identity using integral definition of Hilbert transform

The following is Cotlar's identity, where $H$ is the Hilbert transform $$(Hf)^2=f^2+2H(f\cdot Hf).$$ Applying the Hilbert transform to this identity, we find $$H((Hf)^2)=H(f^2)-2f\cdot Hf \quad(*).$$ ...
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3 votes
1 answer
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Is Dirichlet energy related with entropy?

Intuitively, I feel that Dirichlet energy is related with entropy. And entropy seems to be equivalent with some discrete form of Dirichlet energy. Is this a nice intuition? Is there something worth ...
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1 vote
1 answer
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Fourier transform of a time-variant convolution

For a time-invariant convolution, given the convolution theorem, we know that $\mathcal{F}\{(h*x)(t)\}=\hat{h}(\omega).\hat{x}(\omega)$. My question is what if the convolution is time-variant. Let $(h*...
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6 votes
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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{...
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