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.

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Factorisation of positive definite functions

Let $(S,\circ)$ be a semigroup with identical involution. Of course, we know that products of positive definite functions on $S$ are again positive definite. I'm interested in the other direction, ...
Tobsn's user avatar
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Spherical Harmonics

The spherical harmonics of degree $k$ in $n$ dimensions are the restriction to the sphere $\mathbb S^{n-1}$ of harmonic polynomials homogeneous of degree $k$ in $n$ variables. It is a classical fact ...
Bazin's user avatar
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Existence of an integrable representation

An irreducible continuous unitary representation $\pi$ of $G$ is said to be integrable, if the map $\phi(x)=\langle\pi(x)\zeta,\zeta\rangle$ is integrable on $G$, where that $\zeta\in H(\pi)$. ...
M.fouladi's user avatar
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1 answer
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A geometric proof of the strong maximal theorem

While reading the paper "A geometric proof of the strong maximal theorem", by A. Cordoba and R. Fefferman -Annals of Mathematics Vol 102 no. 1, I got stuck trying to understand a main step in the ...
i like xkcd's user avatar
3 votes
0 answers
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Decomposing a representation of the affine group

Consider the affine group $\operatorname{Aff}(2, \mathbb{R})$ consisting of matrices \begin{equation} A = (g, t) = \begin{bmatrix} g_{11} & g_{12} & t_1 \\ g_{21} & g_{22} & ...
mkreisel's user avatar
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6 votes
1 answer
374 views

Pontriagin reflexivity of the character group

For an Abelian topological group $G$ by $G^{\wedge}$ we denote the Pontryagin dual of $G$, i.e. the group of continuous homomorphisms $G\to\mathbb T:=\{z\in\mathbb C:|z|=1\}$. The group $G^{\wedge}$ ...
Lviv Scottish Book's user avatar
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Reference for the Hardy maximal function on the torus

I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
Ayman Moussa's user avatar
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One question about $L^1(G/K)$ and its closed subalgebra of $K$-invariant functions $L^1(G)^{\sharp}$

Can someone please clarify explicitly why: "The smallest closed subspace of $L^1(G/K)$ containing $L^1(G/K)^{\sharp}$ and invariant under the (left) $G$-action, is the full space $L^1(G/K)$". Where $...
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On a paper by Adams and Frazier

I am reading a paper by Adams and Frazier (namely Adams, Frazier, Composition operators on potential spaces. Proc. Amer. Math. Soc. 114 (1992), no. 1, 155–165, available here), whose main purpose is ...
Mizar's user avatar
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Dimension of roots of irreducible Schur polynomial on unit circle

Let $s_\lambda(x_1,\ldots,x_n)$ be a Schur polynomial in $\mathbb{C}[x_1,\ldots,x_n]$ with $\lambda=(\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n=0)$ and $\gcd(\lambda_1+n-1,\lambda_2+n-2,\dots,\...
M. Hosseini's user avatar
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142 views

(Non-)Existence of certain invariant distributions on a p-adic space

Following Bernstein-Zelevinski, an $\ell$-space is a Hausdorff, locally compact totally disconnected topological space. For an $\ell$-space $X$, denote $S(X)$ the space of Bruhat-Schwartz functions on ...
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Link between subharmonic and subanalytic functions

Consider $\Omega$ an open set of $\mathbb{C}$ and $f : \Omega \to \mathbb{R}$ a $C_{\infty}$-function. The following two definitions are well-known (at least the first one) but I prefer to recall them ...
C. Dubussy's user avatar
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Decay of positive definite function in $L^p$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means $$ \sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i-x_j)y_i y_j \geq 0 $$ for all $...
Linden's user avatar
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Is this averaged exponential sum over primes small infinitely often?

Do there exist infinitely many positive integers $N$ such that $$\sum_{\substack{N/2 \leq q \leq N \\ a/q \notin \mathbb{Z}}} \left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^...
Linden's user avatar
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Need to show bounded behavior of a particular Fourier transform

First let me be briefly state the relevant information to my problem: $\beta(s) \in C_0^{\infty}([-1,1])$, and $\beta \equiv 1$ around $s=0$. The $\beta$ I'm using is an even function, but it doesn't ...
Patch's user avatar
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1 answer
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Proof of the Davies-Gaffney estimate in elliptic pdes?

I'd like a reference to a proof of the Davies-Gaffney estimate; which is an off-diagonal decay result. See for instance assumption H2 in the paper "Hardy Spaces associated to non-negative self-adjoint ...
Lentes's user avatar
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1 answer
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Orthonormal basis and decay

Edit: I added smoothness, hoping to simplify the problem with this additional assumption. Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
Zinkin's user avatar
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1 answer
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A Schur-like product theorem on groups

Let $G$ be a finite group, and consider the composition $X * Y$ on $\mathbb{C}G$ defined by $$(\sum_g \alpha_g u_g) * (\sum_g \beta_g u_g) = \sum_g \alpha_g \beta_g u_g.$$ This composition can be ...
Sebastien Palcoux's user avatar
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131 views

Equivalent statement of the Wiener-Tauberian theorem?

I would like to know why we have the equivalence between the following three statements of the Wiener-Tauberian theorem: version 1: If $I$ is a closed ideal in $L^1(\mathbb R)$, such that the set $...
Z. Alfata's user avatar
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Intersection of thickly syndetic sets

Question: Let $\Gamma$ be a countable group. Is the intersection of two thickly syndetic sets still thickly syndetic? I've only seen the proof for the group $\mathbb{Z}$ (and I believe this method ...
ARG's user avatar
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7 votes
1 answer
469 views

Examples of the large sieve inequality where a constant larger than 1 is needed

Let $S(x) = \sum_{n=0}^{N-1} a_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that $$ \sum_{r=1}^R |S(x_r)|^2 ...
Mark Lewko's user avatar
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5 votes
2 answers
447 views

Fourier support condition in the paper 'A study guide for the $l^2$ decoupling theorem'

I'm currently reading Bourgain and Demeter's study guide for the $l^2$ decoupling theorem (https://arxiv.org/pdf/1604.06032.pdf). I have some trouble with understanding the proof of Proposition 8.4. ...
msaBU's user avatar
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0 answers
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One-dimensional integral equation uniquely solvable?

I recently met a question similar to this one and I would like to post it here, because I basically found nothing: We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
BaoLing's user avatar
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117 views

Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)

How can I prove the following inequality about the Fourier transform? $$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
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A Multiplier Problem for an $L$ Shaped Region

Let $S$ be an $L$ shaped region in the unit cube $Q:=[0,1]\times [0,1]$: $$ S:=Q\backslash C,\quad C:=\left[\frac 1 2,1\right]\times \left[\frac 1 2,1\right]. $$ Consider the multiplier operator $T$ ...
Thomas Yang's user avatar
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1 answer
141 views

Dispersive estimate for linear semigroup

Let's consider the propagator corresponding to the one-dimensional equation $$ u_t=\Lambda^\alpha u_x,\; u(x,0)=f(x) $$ where $$ \widehat{\Lambda^\alpha u}=|\xi|^\alpha\hat{u}(\xi), $$ and $-1< \...
guacho's user avatar
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A kernel on the d-dimensional flat torus with smoothing properties in the $L^{\infty}$-norm

Let $\rho: \mathbb{R}^d\rightarrow \mathbb{R}_+$ be smooth, symmetric, of compact support, and satisfy $\int_{\mathbb{R}^d}\rho(x)dx=1$. For each $\epsilon>0$, set $\rho_{\epsilon}(x)=\epsilon^{-d}\...
user's user avatar
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11 votes
2 answers
2k views

Harmonic function properties on $\mathbb R^3$

Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\...
vibe's user avatar
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4 votes
0 answers
231 views

Fefferman's article: Pointwise convergence of Fourier series, II

I have some problems reading Pointwise convergence of Fourier series by Fefferman https://www.jstor.org/stable/1970917 I got stuck in Chapter 6, Lemma 5. In the proof he split the $\mathcal P'$ into ...
Thomas Yang's user avatar
2 votes
1 answer
997 views

Pointwise convergence implies uniform convergence?

Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like $$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$ Assume that $K\in C^{\text{bounded}...
BaoLing's user avatar
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2 votes
2 answers
67 views

An [IN$]_B$ group with a non-normal compact $B$-invariant subgroup

Let $G$ be a locally compact group with the group of topological group automorphisms $Aut(G)$ furnished with the compact-open topology. Let $B$ be a subgroup of $Aut(G)$. We call $G$ an [IN$]_B$ if ...
Mahmood Al's user avatar
4 votes
0 answers
343 views

Fractional integral inequality (Hardy-Littlewood-Sobolev)

I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
Narek Margaryan's user avatar
1 vote
1 answer
193 views

Does every locally positive-definite function have a positive-definite extension?

Let $B$ denote the unit ball in $\mathbb{R}^d$, and suppose $f\colon B\rightarrow\mathbb{C}$ has the property that for every $n\geq1$ and $x_1,\ldots,x_n\in\mathbb{R}^d$ with $\|x_i-x_j\|<1$, the $...
Dustin G. Mixon's user avatar
4 votes
2 answers
327 views

estimate for a sum of products of Weil's sum

Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define $$ K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)), $$ where $...
Tony B's user avatar
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9 votes
1 answer
527 views

Fefferman's article: Pointwise convergence of Fourier series

I have some problems reading Pointwise convergence of Fourier series by Fefferman https://www.jstor.org/stable/1970917 When I proceed to Lemma 2, Chapter 6, I could not verify either of the following:...
Thomas Yang's user avatar
7 votes
1 answer
453 views

When the value of a function in a point is equal to its integral average over the point's neighborhood?

It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
Grove's user avatar
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3 votes
0 answers
142 views

Is an Abelian topological group compact if it is complete and Bohr-compact?

A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff. A topological group $G$ is Bohr-compact if it admits ...
Taras Banakh's user avatar
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3 votes
0 answers
125 views

An identity of operator norms and de Leeuw's theorem

Let $$Hf(x_1,x_2)=p.v.\int_{-\infty}^\infty f(x_1-t,x_2-S(x_1,x_1-t))\frac{dt}{t},$$ $$T_\lambda f(x)=\lim_{\epsilon\to0}\int_{|x-y|\ge\epsilon}e^{i\lambda S(x,y)}(x-y)^{-1}f(y)dy, $$ where $S(x,y)$ ...
Right's user avatar
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2 votes
1 answer
311 views

Approximation of quasi-periodic function by trigonometric polynomials

The elements of the closure of $\{ \sum_{j=1}^n a_j e^{i\nu_j x}: a_j\in \mathbb{C}, \nu_j\in \mathbb{R} \}$ in the supremum-norm are called almost periodic functions. An almost periodic function $f$ ...
Severin Schraven's user avatar
5 votes
0 answers
152 views

Question about the history of dyadic models in harmonic analysis

Who first used the expression "dyadic model" in the sense of this blog post by Terence Tao? Say you are a harmonic analyst trying to prove a result, e.g., something like the Carleson-Hunt Theorem, ...
Abdelmalek Abdesselam's user avatar
1 vote
0 answers
68 views

When Schroedinger propagator commutes other operators?

Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space). We know that $\widehat{\nabla f}(\xi)= 2 \pi i \xi \hat{f} (\xi). $ We define $$\widehat{|\nabla| f^{s}} (\xi) = (2 \pi |\xi|)^s \hat{f} (\xi), ...
XYZ's user avatar
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1 vote
0 answers
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Hardy $Hp$ norm of similar function

Let $f(z)=\sum_{n=0}^{\infty} \frac{c_n}{n+1}z^n$, where sequence $c_n \in S^1=\{z:|z|=1\}.$ We observe $H^p$ norm $\|f\|_{H_p}$, where $H^p$ is Hardy space, $1 \leq p < \infty$. Question: For the ...
Nebojša Đurić's user avatar
1 vote
0 answers
82 views

Topology of the algebra $\mathbb{C}\{A\}$ for a LCA group $A$

Let $\mathcal{A}$ be a complex associative Hausdorff topological algebra, and let $A\subset\mathcal{A}$ be a locally compact Abelian (LCA) subgroup (multiplicative). The linear span $\mathbb{C}\{A\}$ ...
Bedovlat's user avatar
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7 votes
0 answers
270 views

a question on the paper of Łaba and Wolff

I'm reading the paper A local smoothing estimate in higher dimensions by Izabella Łaba and Thomas Wolff. The paper can be found at J. Anal. Math. 88 (2002), 149–171, doi: 10.1007/BF02786576, arxiv: ...
Tony B's user avatar
  • 443
15 votes
3 answers
855 views

Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $

I am trying to prove or disprove $$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$ where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
Thomas Kojar's user avatar
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3 votes
1 answer
162 views

Uniform bound for an oscillatory sum

I am wondering if there is a uniform bound $C$ (independent of $\lambda>10$): $$\sum_{k=-\infty}^{-1}\Big|\int_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\le C.$$ Remark: (1) An easy upper ...
Right's user avatar
  • 187
2 votes
0 answers
91 views

Spectral multiplier and Littlewood-Paley projection

I am trying to understand this paper, and have some basic question, and hope this is OK for the MO. Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space). We know that $\widehat{\nabla f}(\xi)= 2 \...
XYZ's user avatar
  • 31
2 votes
0 answers
134 views

To find a positive function with compact spectrum

Let $e_1=(0,1)^T$, $$ S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\}, $$ is a cone in $\mathbb{R}^2$. I want to find a non-trivial smooth function ...
John Zhao's user avatar
2 votes
0 answers
60 views

When are solutions of the Schrödinger equation radial?

Let $S$ be a nonnegative self-adjoint operator on a complex Hilbert space $X$. (For example, $X$ consists of functions on $\mathbb R^d$; it could be $L^2(\mathbb R^d), \dot{H}^2(\mathbb R^d)$, etc.)...
abcd's user avatar
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5 votes
0 answers
394 views

Can we prove nowhere differentiability of Brownian path via Karhunen–Loève coefficient?

This post is partly inspired by Fourier Coefficients and Hölder Continuity. Typical proofs of the nowhere differentiability of Brownian paths is by contradiction using binary expansion from real ...
Henry.L's user avatar
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