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, ...

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28 views

Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} ...
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1answer
437 views

Convolution on group with measure

I was wondering about the generalization of the concept of convolution from the familiar one on real spaces and how many properties still remain. For convolution on Lebesgue-integrable real-valued ...
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31 views

Fourier series calculation

I had an exam question today that stated something along the lines of the following: "Let $f$ be an even function given by $f(x)=x$ on $[0,\pi]$ and extend $f$ to $\mathbb{R}$ by $2\pi$-periodicity. ...
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1answer
118 views

Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial ...
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14 views

Laplacian of a radial function on a riemannienne symmetric space

I would like to know : Is that the Laplacian $L$ of any radial function $f$ on a riemannienne symmetric space $X$ is a radial function? Thank you in advance.
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17 views

The decay rate of Hormander lemma is optimal or not?

The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq ...
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5answers
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What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
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2answers
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Proof of the classical div-curl-lemma

let $1 = \frac{1}{p} + \frac{1}{q}$ as usual. Let $f \in L^p, g \in L^q$ be vector fields from $\mathbb R^n$ to itself. Assume $div f = 0$ and there exists a function $G$ s.t. $\nabla G = g$. Then $f ...
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1answer
32 views

Parition of unity argument in a Fourier analysis paper

I am currently reading a paper "The proof of the $\ell^2$ decoupling conjecture" by Bourgain and Demeter. I was wondering if someone could explain me one of the arguments in their paper. First I will ...
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15 views

How does the weighted superposition of irreps make the Fourier transform of a finite group unitary?

This is supplementary to this question. In the lecture note of Andrew Childs on Nonabelian Fourier analysis, it is said that the Fourier transform of a finite group is the weighted superposition of ...
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1answer
34 views

Why is the Fourier transform of a non-Abelian finite group the weighted superposition over all irreps?

I am going through the lecture note of Andrew Childs on Nonabelian Fourier analysis. I would like to quote from the note: My question: Why does it have to be weighted superposition and not equal ...
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1answer
30 views

On a property of $(Mf)^{\delta}$

For each positive $C$, define a set $$A_C=\left\{g\ge0: \frac{1}{|I|}\int_Ig\le C\inf_{x\in I}g(x) \text{ for any interval } I\right\}$$ In other words, elements in $A_C$ are non-negative functions ...
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1answer
48 views

The support of $f(x)= \cos(x)$

The support of a function is the closure of the set of points where the function has non zero values. The function $f(x)=\cos(x)$ is zero only at the points $x=\frac{(2k+1)\pi}{2}$, $k \in ...
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1answer
47 views

Properties of the Fejer kernel

The Fejer kernel $k_m : \mathbb{R} \to \mathbb{C}$ is defined by $k_m (t) = \frac{1}{2\pi (m+1)} \sum^m_{n=0} \sum^n _{k=-n} e^{ikt}$ One of the properties of the Fejer kernel is For any $\delta ...
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0answers
35 views

Strong period of self-convolution of a strongly periodic cyclic function

Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all $j \in ...
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1answer
98 views

Inverse short time Fourier transform

The short time Fourier transform $S: L^2(\mathbb{R})^2 \rightarrow L^2(\mathbb{R}^2)$ can be defined as $$S(g,f)(a,b):=\int_{\mathbb{R}}f(x) \overline{g(x-a)} e^{-i b x} dx.$$ Now a natural question ...
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43 views

Interpretation of $L_2 ([-\pi , \pi])$

What is the interpretation of $L_2 ([-\pi , \pi])$ in laymans terms? How is $L_2 ([-\pi , \pi])$ different to $L_1 ([-\pi , \pi])$? Does $L_2 ([-\pi , \pi])$ just mean that the function is twice ...
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1answer
41 views

Interpolation of a linear operator acting on a sequence of functions

Let $\mathbf{f} = \{f_{n}\}$ be a sequence of Schwarz functions and suppose $T$ is a linear operator which sends a given sequence of Schwarz functions to a given function in $L^{p}(\mathbb{R}^n)$ for ...
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1answer
40 views

Is $\|\widehat{f}\|_{L^{p}(\mathbb{R}^d)}$ comparable to $\|\widehat{f}\|_{L^{p}(B_{1/R})}$?

Suppose $f$ is a smooth function compactly supported in some ball of radius $R$. Is $\|\widehat{f}\|_{L^{p}(\mathbb{R}^d)}$ comparable to $\|\widehat{f}\|_{L^{p}(B_{1/R})}$ where $B_{1/R}$ is any ball ...
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2answers
180 views

average of maximal function is less than its infimum?

Let M be the dyadic Hardy-Littlewood maximal operator. Prove the following: there is a constant $C$ such that for any $f$, $$ \inf_{x\in I}Mf(x)\le C 2^k\inf_{x\in J} Mf(x) $$ where $I$ and $J$ are ...
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2answers
32 views

Showing $\hat{\tilde{f}}=\tilde{\hat{f}}$ where $\hat{f}$ is the Fourier transform, and $\tilde{f}(x) = f(-x)$

I'm trying to prove that $\hat{\tilde{f}}=\tilde{\hat{f}}$ for any integrable function $f$, where $\hat{f}$ denotes the Fourier transform of $f$ and $\tilde{f}$ denotes the mapping $f(x)\to f(-x)$, ...
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1answer
66 views

Question regarding constructing a function via its Fourier transform

Let $\varepsilon>0$. I was interested in understanding the justification of defining the following function $\phi$ via its Fourier transform, satisfying the following properties: (1) ...
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30 views

Proof of an inequality with using maximal operator

I want to prove an inequality such that $$ \int_{B}|f(y)|dy\leq |B|^{1-\frac{1}{p}}\|f\|_{L^p(B)}, $$ where $B\subset\mathbb{R}^n$ is a ball, $p>1$ and ...
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34 views

Question related to decay of Fourier transform and smoothness

Suppose $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ Let $$g(x) = \frac ...
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2answers
81 views

$W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : ...
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0answers
37 views

Additive combinatorics in Hamming weights of addition of numbers modulo $2^n$ with prescribed Hamming weight

I wonder if anyone could point me to a reference about the following type of combinatorics problem: Fix $n $. For an integer $k \in [n] = \{1, \ldots, n\}$, let $A(k)$ be the set of integers in $[0, ...
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1answer
15 views

Proving Holders inequality for the sequence space $l_p (\mathbb(N)$

We first look at when $p=1$ and $q=\infty$ And we look at the non trivial case when the sequences $x=(x_k)_{k \in \mathbb{N}}$ and $y=(y_k)_{k \in \mathbb{N}}$ are both not equal to zero. We first ...
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44 views

How is $\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$?

I'm looking at some material on harmonic functions and it says $$\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$$ where $n$ is the number of dimensions and $\omega_n$ ...
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1answer
48 views

Convolution of indicator functions with values in a finite field

This is something I haven't seen online yet, indicator functions with values in a finite field. Probably for a good reason, but I would like to know why, and if there are still things that can be ...
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1answer
38 views

Can the system of shifts of an $L^2(\mathbb{R})$ function be an ONB?

In Wavelet theory, one constructs wavelet bases via translations a dialations of an $L^2$ function... Is it possible for some set of translations alone to form an Orthonormal Basis? That is: Does ...
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1answer
50 views

Critical Homogeneous Sobolev Embedding

For $s\in\mathbb{R}$, $1<p<\infty$, define the homogeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^{n})$ as follows: For $f\in\mathcal{S}_{0}(\mathbb{R}^{n})$ (Schwartz functions with Fourier ...
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0answers
35 views

Real zeros of a function holomorphic in the upper half plane

Suppose that $f(z)$ is holomorphic in the upper half-plane and $\lim_{\, |z| \to \infty} f(z) = 1$ along any ray in the open upper half-plane. Suppose also that $f(z)$ has a continuous extension to ...
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1answer
50 views

Finding a continuous $f\in C(\mathbb{T})$ with $\{\hat{f}(n)\}\notin \ell^p$

Denote the circle by $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Find an example for a continuous function $f\in C(\mathbb{T})$ with coefficients of the fourier series, $\{\hat{f}(n)\}\notin\ell^p,\forall ...
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24 views

Are exponential functions a complete system on $L^2\left(\mathbb{T},\mu\right)$

Let $\mathbb{T}$ denote the unit circle in $\mathbb{C}$, and let $\mu$ be a complex Borel measure on $\mathbb{T}$. Consider the space of square integrable functions $L^2\left(\mathbb{T},\mu\right)$. ...
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1answer
26 views

Convergence of measures on $\mathbb{T}$

Denote by $M(\mathbb T)$ the set of complex-values measures on the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$.Prove that $D(T)$, the set of discrete measures on $\mathbb{T}$ is: closed in ...
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1answer
106 views

Convolution algebra $L^1(G)$ for non sigma-finite $G$

Let's assume that $G$ is locally compact and Hausdorff topological group, hence it carries a Haar measure, $\mu$. We can than consider space of integrable functions $L^1(G)$ (class of functions to be ...
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1answer
19 views

Convergence in $H^1(\Omega)$ and $L^2(\Omega)$

Let $\Omega$ be a bounded domain (maybe that doesn't matter), if $f_n\rightarrow f$ in $H^1(\Omega)$, does it follow $f_n\rightarrow f$ in $L^2(\Omega)$ since $H^1$ is dense in $L^2$? Is it true that ...
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1answer
41 views

Example of a compactly supported Lipschitz function with non-Lipschitz Hilbert transform

Suppose $f$ is a compactly supported measurable function (say in the interval $[-1,1]$) which is Hölder continuous of order $\alpha\in (0,1)$. I have read that the Hilbert transform $Hf$ of $f$ is ...
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21 views

Spectrum of Laplacian on divergenceless vector on $T^3$

I encountered the following calculation: $\int dA A_\mu\Delta A^\mu$, where $A_\mu$ are divergenceless vectors and the theory is on flat $T^3$. What is the determinant we get by integrating over ...
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1answer
24 views

Why such a net will exist?

G- locally compact group & $\lambda(x)f(y) = f(x^{-1}y)\ \forall \ y \in G$. The following condition is called Reiter's finite condition. $P(G) := \{f \in L^1(G): f \geq 0, \|{f}\|_1 = 1\}.$ ...
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32 views

Boundedness of singular integral operators on $L^{p}$ spaces

Let $\Omega \in L^{1}(S^{d-1})$ have mean zero. Prove that, if the operator $T_{\Omega}: L^{p} \rightarrow L^{q}$ given by $T_{\Omega}f(x) $:= p.v. $\int_{\mathbb{R}^{d}} \frac{\Omega ...
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1answer
36 views

Why a left-invariant Haar measure is $\mu(A)=\int_A \frac{1}{a^2}da\,db$ and a right-invariant Haar measure is $\mu'(A)=\int_A\frac{1}{a}da\,db$?

Let $G$ be the group of affine transformations of $\mathbb R$, $x\mapsto ax+b$, $a>0$. $G$ is the half-plane $(a,b);a>0$. A left-invariant Haar measure is $\mu(A)=\int_A \frac{1}{a^2}da\,db$, ...
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43 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
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0answers
20 views

Spherical resolvent kernel on $H^n(\mathbb R)$

Is there an explicit formula in the literature for the spherical resolvent kernel $R_{\lambda}(r)$ of the Laplacian $\Delta_{H^n}$ on $H^n(\mathbb R)$ the real hyperbolic space ? Such that: ...
4
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0answers
58 views

Fractional Sobolev Space Trace Inequality

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and ...
4
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1answer
43 views

Fundamental solution of a shifted operator

what is the fundamental solution of the shifted operator $ \Delta + \lambda^2 $, i.e, what the function $f$ satisfying the following equation $$ (\Delta + \lambda^2 )f(x) = \delta(x),$$ where $ \Delta ...
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1answer
33 views

What condition on $f$ makes the formula $(−\Delta)^sf(x)=c_{n,s}\int_{\mathbb{R}^n}\frac{f(x)−f(y)}{|x−y|^{n+2s}}dy$ true?

I'm trying to understand the concept of fractional Laplacian, and I found the page https://www.ma.utexas.edu/mediawiki/index.php/Fractional_Laplacian,and the formula ...
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0answers
10 views

Resolvent kernel of Hyperbolic space

The expression of the spherical functions $\varphi_\lambda(x)$ and the resolvent kernel $r_\lambda(x)$ on Hyperbolic space are known. The spherical functions are the Jacobi functions ...
2
votes
1answer
41 views

Hardy-Littlewood theorem about the Poisson integral for $p=1$

(Hardy-Littlewood Theorem) : Let ‎$ u(r,‎\theta)‎$‎ be the Poisson integral of ‎$ ‎\varphi ‎\in L‎^{p}‎‎$‎, ‎$ 1<P ‎\leqslant‎ ‎\infty‎$‎ , and let $ U(‎\theta)=\sup‎_{r<1}‎|u(r,‎\theta)|‎ $‎. ...
2
votes
2answers
80 views

Product of two sinusoidal functions model

I'm trying to make a model of the rise and fall of sea levels. According to this explanation and image in the textbook, the product of two sinusoidal functions should look something like this: ...