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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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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How come $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ has the form of $e^{-in\phi_i}$ and how to calculate it's eigenvalues

How to prove Euler's Formula $e^{i\theta} = (cos\theta + isin\theta)$? I know this is kind of basic and I am familiar with this equality for a long time. But, how do I prove it? And another ...
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151 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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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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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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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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36 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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31 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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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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35 views

Gradient Estimate for Logarithmic Cutoff Function

In Notes 4 on regularity of harmonic maps, T. Tao asserts the following lemma, which in short, allows us to localize a map $u:\Omega\rightarrow S^{m-1}$ (here, $\Omega\subset\mathbb{R}^{2}$ is a ...
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36 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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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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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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100 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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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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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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33 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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33 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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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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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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17 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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33 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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1answer
49 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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83 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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39 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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36 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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51 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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25 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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51 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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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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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
20 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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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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34 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
25 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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1answer
39 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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108 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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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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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: ...
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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 ...
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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 ...
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42 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)|‎ $‎. ...
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61 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
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108 views

Is Rudin correct here? Fubini's theorem and product measures

Let $X, Y$ be locally compact Hausdorff spaces with nonnegative regular measures $\mu, \lambda$. By definition (in the book I'm reading) a regular measure is a Borel measure for which every Borel set ...
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87 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: ...
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33 views

Fourier Transform of a kernel

Let $\omega \in \mathcal{S}(\mathbb{R}^{2})$ and define $u(x) = \int_{\mathbb{R}^{2}}\frac{(x-y)^{\perp}}{|x - y|^{2}}\omega(y)dy$, where $(x-y)^{\perp} = \begin{bmatrix}x_{2} - y_{2}\\-x_{1} + ...
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85 views

Continuous Littlewood-Paley Inequality

I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4): For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates ...
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18 views

Exactness of harmonic sum using Mellin transformation

I'm trying to learn how you can use the Mellin transformation to obtain closed expressions of harmonic sums. There are demonstrations om MSE how show this technique. eg Proving $\sum_{n ...
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1answer
41 views

Real vs Complex Interpolation

The two major classical interpolation theorems in analysis are Riesz-Thorin Theorem (complex method) and Marcinkiewicz Theorem (real method). One can see the statements of the theorems and realize ...