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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Imaginary complex numbers

Let $z=x+iy$ and $v=2xy$, show that $v=Im[z^2]$ and find a harmonic conjugate of $v$ on domain $D$. Also find the largest domain $D$ on which $v$ is harmonic.
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Fourier algebra and multipliers of a finite discrete group

Let $G$ be a finite discrete group. We denote by $A(G)$ the Fourier algebra of $G$ and $M_{cb}A(G)$ the space of completely bounded multipliers of $A(G)$. Is is true that $A(G)=M_{cb}A(G)$ ...
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136 views

Which spherical harmonic function will correspond to such a representation?

On Wiki there's a figure displaying "visual representations of the first few spherical harmonics." I was wondering exactly which spherical harmonic function will generate a representation like this ...
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174 views

Another aspect of Heisenberg uncertainty principle

In fourier transformation theory, we have the Heisenberg uncertainty principle, i.e. Suppose $\phi$ is a function in [Schwarz space][1] which satisfies the normalizing condition ...
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Recovering a group from its C*-algebras and group algebra

Let $G$ and $H$ be locally compact groups. Does anyone know the answers to these questions? Is it true that: if $C^*(G)$ and $C^*(H)$ are $*$-isomorphic, then $G\cong H$? if $C_r^*(G)$ and ...
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How to show that $ν(z)$ is Carleson measure

If $ν(z)=|1+z|^ β dμ(z)$, $β\in \!\ R^-$. How to show that $ν(z)$ is Carleson? I know that $ν$ is Carleson measure if $ν(Q_I)\leqq \!\ c.I$ But how to apply this?
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Harmonic function, existence of a constant

May i ask you for a little help about a problem with harmonic function? It seems to be not that difficult, in a way even intiutively obvious but i don't really know how to show this explicitly. We ...
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242 views

Are nilpotent Lie groups unimodular?

The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by \begin{equation*} \int_G f(xy)dx = \Delta(y)\int_Gf(x)dx \end{equation*} where $dx$ is a left Haar measure on ...
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51 views

$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq \int_{-\infty}^{\infty}|f|^2 dx$?

Suppose complex function $f$ in the Schwartz Space, its definition see http://en.wikipedia.org/wiki/Schwartz_space how can we argue that $$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq ...
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27 views

$(y^2+2iy)^{m-\frac{1}{2}}=y^{m-\frac{1}{2}}(2i)^{m-\frac{1}{2}}+O(y^{m+\frac{1}{2}})(0\leq y \leq 1)?$

This question comes from stein's book Introduction to fourier analysis on euclidean space,page 159. when $m > 1/2$, ...
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532 views

A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
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186 views

Orthonormal basis in Hilbert spaces

I have a general question but I'm going got ask it in a very restrictive setup. It is known that an equivalent condition for a system $\left\{e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ being an ONB ...
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791 views

About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$

How to prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$ $H_n$ is the n th harmonic number
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Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in ...
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1answer
252 views

Fourier transform of a function is square integrable

Is there a result stating that if a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable and decays at infinity, then its Fourier transform is also square integrable?
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44 views

$L$ elliptic diff op $\implies$ singsupp$(u)\subseteq$singsupp$(Lu)$ for distributions $u$?

If $L:D'(\mathbb{R}^n)\to D'(\mathbb{R}^n),n\in\mathbb{N}$ is a weakly elliptic, linear differential operator with constant coefficients then for every $\Omega\subseteq\mathbb{R}^n$, and for all $u\in ...
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440 views

Prove partial derivatives of uniformly convergent harmonic functions converge to the partial derivative of the limit of the sequence.

I think the title says it all. If you have a sequence of harmonic functions from a bounded complex domain to the real numbers, show that on a subset at a positive distance from the boundary of the ...
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25 views

$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...
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200 views

Proof of Euler's general formula for a sum involving harmonic numbers [duplicate]

I have seen this formula, but how to prove this? $$2\sum\limits_{k=1}^\infty \frac{H_k}{\left( k+1 \right)^m} =m\zeta \left( m+1 \right)-\sum\limits_{k=1}^{m-2}{\zeta \left( m-k \right)\zeta \left( ...
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48 views

Variation on a completeness relation

The completeness relation for the spherical harmonics is: $$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
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1answer
41 views

A Quadratic Maximum?

What does the following mean? Context: Laplace integrals Consider the integral $$\int_a^b f(t) \exp(x\phi(t))\,\,\,dt$$ where $\phi(t) $ achieves its maximum at some $t=c\in (a,b)$. Assume that ...
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204 views

Help proving Calderón reproducing formula (simple version)

Let $\phi$ be a real compactly supported smooth function on $\mathbb R$ with total integral zero. Define $\phi_t=\frac{1}{t} \phi(\frac{x}{t})$. I also suspect that they must be even, but the notes I ...
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167 views

Is the zero set of a non zero real valued harmonic function discrete?

It is a basic fact that the zero set of a non zero holomorphic function defined on a open set $A$ is discrete. By a result in Rudin's textbook on "Real and Complex Analysis", we know that any real ...
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How to calculate HarmonicNumber(x, 1.6)?

I got this solution while using Wolframalpha: $$ (-1.66667 n+1/2-0.133333/n+0.0208/n^3-0.0127573/n^5+O((1/n)^6))/n^1.6+2.28577 $$ Could somebody tell me this solution step-by-step? I need to write ...
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475 views

Hardy-Littlewood-Sobolev inequality for $p=1$

Let $\mu $ be a positive Borel measure on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some $n\in (0,d]$ ...
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314 views

$1/|x|^n$ is not integrable

Let $\mu $ be a positive Borel measure on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some $n\in (0,d]$ ...
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261 views

Simply connected domain and harmonic function

Let $\Omega$ be a simply connected domain that is properly contained in $\mathbb C$, and $u(x,y)$ is harmonic on the unit disk $\mathbb D $, then there is a funtion $f(z)$, that is one-one and ...
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A question from Stein's book, Singular Integral.

A question from Stein's book, Singular Integral. Let $\left\{ f_{m}\right\} $ be a sequence of integrable function such that $$\int_{% \mathbb{R}^{d}}\left\vert f_{m}\left( y\right) \right\vert dy=1$$ ...
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Laplace equation Dirichlet problem on punctured unit ball.

Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem \begin{align} \Delta u &= 0 \\ u(0) &= 1 \\ u &= 0 ~~~\text{if} ~~|x|=1 \end{align} By considering ...
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208 views

non tangential maximal function and Hardy-Littlewood maximal function

I'm studying harmonic analysis and found that we can bound non-tangential maximal function by Hardy-Littlewood maximal function. Most books don't give the proof of it. How can I see that? Is there a ...
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74 views

Simple Harmonic estimate

I know this is simple, and I see all the pieces of the puzzle are there, but I can't seem to get it. Let $u$ be a solution of $$\Delta u = f \;\;\; x \in B_4 $$ Then if we can bound $$\int_{B_4} ...
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351 views

How to evaluate integrals with respect to Lebesgue measure on the unit sphere?

Let $\sigma$ be the "normalized Lebesgue" (Haar, really...) measure on the unit sphere $S=S^{n-1} \subset \mathbb R^n$. That is, $\sigma$ has support $S$, it is uniformly distributed, and $\int_S ...
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253 views

Decaying Fourier transform and smoothness

Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ I want to show ...
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103 views

Continuous, integrable fourier transform of an $L^{2}(\mathbb{R})$ function, integrable.

I've come across a number of sources claiming a smoothness-decay duality between a function and its Fourier transform. But most seem to give results about how the smoothness of a function leads to ...
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1answer
255 views

Convolution square root of $\delta $

I want to somehow classify the distributional solutions of the equation $$ f \ast f = \delta $$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
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111 views

Is this function a subharmonic function?

Does anyone know, is $h\left(z,w\right):=\frac{\left|zw\right|}{\left|z\right|+\left|w\right|}$ for $z$ and $w$ in the unit disk $\mathbb{D}$ of the complex plane a (pluri)subharmonic function? ...
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197 views

Dirichlet Problem: Uniqueness of solution

Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$. Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$? ...
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135 views

Canonical field of Hilbert spaces in Dixmier; Plancherel Theorem

I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof ...
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1answer
92 views

Dirichlet problem: Obtaining the harmonic measure through Riesz representation theorem

For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$ $$ \Delta u=0, \text{ on } D, \\ \left. u\right|_{\partial D}=f \in C\left( \partial D\right). $$ With a fix $x$ in $D$, an ...
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132 views

Dirichlet problem: Is the Poisson Integral always a solution?

Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$. Is the Poisson integral of $f$, $$ Pf(x)=\int_{\partial D} f(t) ...
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How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
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Alternate definitions of $C^{1,\alpha}(S^1)$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definitition regarding the following: Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
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Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$

I'm trying to learn the Selberg trace formula, but have very little background. I was referred to Dennis Hejhal's The Selberg Trace Formula for $PSL(2, \mathbb{R})$ but just got the book and was ...
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267 views

Harmonic function with condition on part of its boundary

Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...
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235 views

Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$

Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which $$ \lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v. $$ But ...
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Suggestion for a project on Harmonic measure and Fourier analysis

I have a course project on harmonic measure and Fourier analysis. The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis. Harmonic measure is a vast ...
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231 views

How does Pontryagin duality fit into the general cohomology theory framework?

Pontryagin duality implies the isomorphic relation of the function space $C(G)$ on a locally compact group $G$ to the function space on it's dual group $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
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94 views

Proof for Fourier transform in $L^2$

This question makes me really confused: Let $f$ and $g$ two functions in $L^2$. Show that: $$\int \widehat f\cdot gdx= \int f\cdot\widehat gdx,$$ where $\widehat f$ is the Fourier transform ...
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104 views

The various central extensions of $(G\times G)$ by $T$.

Let $G$ be a locally compact abelian group, isomorphic to $G^*$, its Pontryagin Dual. Let $T$ denote the unit circle in $\mathbb{C}$, where continuous morphisms $\chi: G\to T$ are the elements of ...
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233 views

Radial limits of harmonic conjugate and Hilbert transform

Let $\mu$ be a real measure on the circle $\mathbf{T}$. Then the function $$f(z)=\int_\mathbf{T} \mathrm{Im}\left(\frac{\zeta+z}{\zeta-z}\right) d\mu(\zeta)$$ is harmonic on the unit disc and its ...