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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construction of a partition function

Let $q$ be a large positive integer. How to construct a smooth function $\phi$ with the following properties? i) $\sum_{a\in\mathbb{Z}}\phi(q(x-a/q))=1$ for any $x\in\mathbb{R}$ ii) For any ...
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Integration over ideles over $\Bbb{Q}$ , Tate`s thesis special case

Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of ...
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Tate thesis : Global functional equation [closed]

It will be very helpful if someone tells me how to do EXERCISE 1 here. I have done part $2$. I cannot do part $1$ and part $3$. I tried part $1$ by decomposing $Z(f,s)$ as the product of its local ...
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Does band-limited imply continuous?

I was hoping someone could point me to an article or text which explores the connection between the continuity of a signal in the time domain and it being band-limited in frequency domain. (Update) ...
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108 views

The regularity of elliptic equation

Define $$ Lu:=-\partial_j(a_{ij}\partial_iu)+b_i\partial_iu+cu $$ where $a_{ij}$, $b_i$, and $c$ are all of $C^\infty(\bar \Omega)$ and we assume $\Omega$ is open bounded in $R^N$, smooth boundary, ...
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In the real spherical harmonics, where does the sqrt(2) factor come from?

The real spherical harmonics can be written in terms of the complex spherical harmonics: $$ Y_{\ell m} = \begin{cases} \displaystyle \sqrt{2} \, (-1)^m \, \operatorname{Im}[{Y_\ell^{|m|}}] & ...
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1answer
31 views

Calculating derivative of a radial function: am I doing it right?

Let $f: \mathbb R^n \setminus \{0\} \to \mathbb R^n \setminus \{0\}$ be such that $f$ only depends on the distance from the origin, that is, $f=f(r)$ where $r = \sqrt{\sum_{i=1}^n x_i^2}$. I am ...
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“Tube domain over symmetric cone” thesis references?

My professor has told me to read "tube domain over symmetric cone". While saying so he said something related to complexificatin of real Lie algebra and representation theory. What is connection ...
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75 views

Limit of bounded harmonic functions is harmonic

I am trying to solve this old qual problem: Suppose $\{u_n\}_{n=1}^\infty$ is a sequence of functions harmonic on an open set $U \subset \mathbb{C}$ and uniformly bounded by 1. Suppose there is a ...
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55 views

A problem for laplace operator in Sobolev space

Suppose $u\in L^2(\Omega)$, then for any $\phi\in C_c^\infty(\Omega)$ we have $$ \int_\Omega v\,\phi\,dx=\int_\Omega u\Delta \phi\,dx $$ Then can I conclude that $u\in H_0^1\cap H^2(\Omega)$ and ...
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If $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well.

I know the theorem that if $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well. The book I know that contains the proof of this theorem all use the approach with respect to complex ...
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133 views

Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
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Using taylor formula to compute laplace operator

Suppose $u\in C^2(\Omega)$ and $x\in \Omega\subset \mathbb R^N$. I am trying to prove that $$ \Delta u(x)=\lim_{r\to 0} \frac{2N}{r^2} \left[\frac{1}{\alpha(N)} \int_{\partial ...
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Group C*-algebra of an abelian discrete group

Let $A = \mathbb C[G]$ be the group ring of all finitely supported functions $f\colon G \to \mathbb C$ of a discrete abelian group $G$ with the usual convolution product, and involution defined by ...
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71 views

Decomposition and harmonic analysis of $L^2(S^n)$

Deitmar and Echterhoff write in their book Principles of Harmonic Analysis that `It follows from the Peter-Weyl theorem that the $SU(2)$ representation on $L^2(S^3)$ is isomorphic to the orthogonal ...
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43 views

relationship between BMO norm and $L_p$ norm

Are there any relationship between the BMO norm of a function and its $L_p$ norms? For example, one norm is controlled by the other for functions of some special class.
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138 views

Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
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75 views

Sublinearity of Hardy-Littlewood Maximal Function on Sobolev Spaces

Define the (centered) Hardy-Littlewood maximal function by $$\mathcal{M}f(x)=\sup_{r>0}\dfrac{1}{m(B(x,r))}\int_{B(x,r)}\left|f(y)\right|dy,\ f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$$ We say ...
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41 views

Finding an example

I am looking for an example to have two distinct decreasing functions $g_1$ and $g_2$ between [0,1], such that their ratio ($\phi=\frac{g_1}{g_2}$) is a periodic function with period one, moreover for ...
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76 views

The Cauchy problem for Laplace equation in unit cube

Here is the question: We are given a laplace equation $\Delta u=0$ in $Q:=(0,1)\times(0,1)$. Q1: What are some conditions you can put on this to get uniqueness/existence? Q2:What if you wanted to ...
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Decomposition of regular representation

Let $G$ be a compact group. Then there is an isomorphism $L^2(G)\simeq \bigoplus_{\tau\in \hat{G}} V_{\tau}\otimes V_{\tau^*}$ which intertwines the conjugation action of $G\times G$ on $L^2(G)$ ...
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155 views

The well-posedness of Laplace equation on half-space

In $2$ dimension, take $-\Delta u=0$ on $\{(x,y\},y\geq 0\}$ with $u(x,y=0)=f(x)$, $u_y(x,y=0)=g(x)$ where $f$ and $g$ are smooth function. I want to justify whether this problem is well posed. My ...
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494 views

Neumann problem for Laplace equation on Balls by using Green function

It is well known that for Dirichlet problem for Laplace equation on balls or half-space, we could use green function to construct a solution based on the boundary data. For instance, one could find a ...
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Functions over a finite domain that cannot be represented by Fourier series

Given a double Fourier series for some $f:[0,L]\times [0,R]\to \mathbb{R}$ of the following form $$\sum_{k,l=0}^{\infty}a_{kl}\cos\left(\frac{2k\pi ...
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66 views

Jump discontinuity of a function and its analytic phase

Let $f:\mathbb{R}\to \mathbb{R}$ and $f \in BV(0,1)$ with a jump discontinuity at $x = a\in(0,1)$. Let $f_h$ be its Hilbert transform and let $$f_A(x) = f(x) + i f_h(x)$$ Is it true that the function ...
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53 views

A nonlinear Poisson equation problem. (Related to Laplace equation)

Let $\Omega\subset R^N$ be open bounded. Define \begin{cases} \Delta u = f(u) &x\in\Omega\\ u=1 & x\in\partial \Omega \end{cases} Q1: Suppose $f(u)=u^m$ where $m$ is odd. Prove that if there ...
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41 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
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47 views

Left and Right Discrete Maximal Functions

Define the uncentered maximal function $$\widetilde{M}f(n)=\sup_{s,r\in\mathbb{Z}^{+}}\dfrac{1}{s+r+1}\sum_{k=-r}^{k=s}\left|f(n+k)\right|,$$ where $\mathbb{Z}^{+}=\left\{0,1,2,\ldots\right\}$. Define ...
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32 views

Restriction estimates

What is the defining property of what someone in the harmonic analysis community would call a "restriction estimate?" I see sobolev norms, Fourier transforms, and inequalities relating these. The ...
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Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
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149 views

Is there a “unifying framework” for harmonic analysis?

Recently, I was exposed to a basic harmonic analysis course. Although the course is almost over, I still can't put my finger on what harmonic analysis is about. I have a vague idea that it is ...
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Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
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72 views

Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ ...
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1answer
34 views

Time series closed form similar to harmonic series

I want to find the closed form for the following: $$ S(k, \alpha) = \sum_{t=T-k}^T \frac{\alpha^{-t}}{t} $$ when $\alpha \in (0,1)$. For harmonic series there is an easy way to upperbound: $$ H(k) ...
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An $f\in H^{1/2}$ with self-convolution, showing it is an $C^1$ function.

If $f\in H^{\frac{1}{2}}(\mathbb{R})$ is a Sobelev 1/2 function that $f=f*f$, then how do you show that $f\in C^1$ with a bounded derivative.
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Showing $f$ is an $L^p$ function if $f$ is "self-convoluted.

If $f$ is $L^2(\mathbb{R})$ and $f=f*f$, show that $f$ is $L^p$ for $2\leq p\leq \infty$.
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28 views

Plancherel's theorem variants

How would you prove a variant form of Plancherel theorem: If $(c_n)_{n\in\mathbb{Z}}$ are coefficients and $\sum_{n\in\mathbb{Z}}|c_n|^2<\infty$, then there exists a unique function $g\in L^2(0,1)$ ...
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Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
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405 views

Hardy–Littlewood-Sobolev inequality without Marcinkiewicz interpolation?

Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| ...
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104 views

Big theta notation of harmonic series

I want to prove that big theta notation of the harmonic series is $\Theta(\log n)$. I want to work with integral to show that. I attempted this: $$\ln(n)=\int^n_1 \frac{dx}x \le \sum _{k=1} ^n ...
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Orthonormal system

Let $\varphi\in L^2(\mathbb{R})$, prove that $\{e^{2\pi i m x}\varphi(x)\}$ is an orthonormal system iff $$\sum_{n\in\mathbb{Z}}|\varphi(x-n)|^2=1 \ \ a.e \ x$$ How do you prove this. The hint is ...
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Can a “tangent vector of a discrete group” be extended to a tangent vector of its $C^*$-algebra?

This is related to my recent question in MO. I am sure this is trivial, but I have no intuition here, so my apologies from the very beginning. Let $G$ be a discrete group, $A$ a $C^*$-algebra, and ...
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791 views

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB?

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB? I just need help in confirming if my way of approach to finding the THD seems valid, I'm new to MATLAB so I'm not quite ...
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25 views

Connections of results in Harmonic analysis in the theory of Transcendental Numbers

Note :This question is proposed 2 years ago in MO , I see it appropriate for stackexhange math, i posted it here as it's unsolved problem and has a connection with Transcendental Numbers , mayeb we ...
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How to construct $\operatorname{End}(V_{\pi})$ using a representation $\pi$

Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning. By the very definition, for ...
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Estimation of trignometric polynomial and Lipschitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
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75 views

Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
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83 views

Harmonic analysis on discrete groups

I am currently confused about the notion of generalized Fourier transforms on discrete groups (I'm only concerned about discrete groups here, not necessarily abelian). On finite discrete groups, I ...
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61 views

Why can real variable methods take place of complex variable methods in harmonic analysis' research?

We know that one complex variable methods have been largely used in the early research of (real variable) harmonic analysis. But, as is known to me, there is much difficulty when mathematicians ...