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- Lebesgue space and Fourier transform

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
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Is Fourier transform of a $L^{1}$ integrable function is $L^{1}$ integrable?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi $ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. Let ...
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Convolution of distribution and Poisson kernel

I know that for a general tempered distribution (see here) $f$ the convolution $f\star P_t$ is not meaningful. Where $P_t$ is the Poisson kernel (see here) which is given by ...
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How to estimate (compute) Fourier transform?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi $ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. ...
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Compute $\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right )^2+\left (\frac{1}{n} \right )^2$

Compute the value of the following expression $$\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\left ( \frac{1}{2}+\cdots + \frac{1}{n}\right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right ...
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180 views

Taylor expansions of harmonic functions.

Let $D \subset \mathbb{R}^2$ be the unit open disc. Note that any harmonic function on $D$ is real analytic. How can one prove that there exits a constant $C>0$ such that the Taylor expansion of ...
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70 views

Higher Dimensional Paley Wiener Theorem

Is there a natural generalization of the Paley Wiener theorem to higher dimensions (i.e. relating a function $f \in L^2(K), K \subset \mathbb{R}^d$ compact, to an entire function in $\mathbb{C}^d$)? ...
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47 views

how to deduce that finite group $G$ has at most $|G|$ characters

Let $G$ be a finite abelian group. A character of $G$ is a group homomorphism $\chi: G\longrightarrow \mathbb{C}^{\times}$. I have proved by induction that for distinct $\chi_1,\cdots,\chi_r$, they ...
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A bound for the product of two functions in BMO

The question: Let's consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I'm trying to figure out if the following inequality is true $$ \|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}. $$ ...
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Show the following: $\displaystyle\sum_{n=1}^\infty\dfrac1{n(n+k)}=\dfrac{H_k}k$.

For each $n\in\Bbb N$ with $n\geq1$ is $\displaystyle H_n:=\sum_{k=1} ^n\dfrac 1k$ the $n$*-th partial sum of the harmonic series.* $k\in\Bbb N$ with $k\geq1$. Show that ...
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58 views

Questions about Haar integral.

Questions about Haar integral. Let $B$ be the subgroup of $GL_2 (\mathbb{R})$ defined as $$ B =\{ \left( \begin{matrix} 1 & b \\ 0 & c \end{matrix} \right), b, c \in \mathbb{R}, c \neq 0 ...
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31 views

Questions about Haar integral for the group $GL_2(\mathbb{R})$.

Questions about Haar integral for the group $GL_2(\mathbb{R})$. How to show that a Haar integral for the group $GL_2 (\mathbb{R})$ is given by $$ I(f ) = \int_{\mathbb{R}} \int_{\mathbb{R}} ...
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1answer
118 views

Dirichlet triangle mesh

I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My ...
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73 views

A question on Harmonic functions.

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} ...
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(Obvious?) Half-Space Poisson Kernel Estimate

In Stein's Singular Integrals and Differentiability Properties of Functions, Theorem 1 (a) of Chapter 8 (pg. 197) he makes the claim without proof that the Poisson Kernel for the half-space ...
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for any $k\in N$, p.v$\int_{a}^{b}\frac{\cos kx}{t-x}dx=?$

We know that the Hilbert transform of cosine function is sine,see http://mathworld.wolfram.com/HilbertTransform.html. Now, we don't integral from $-\infty \to \infty$. We integral from $a \to ...
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Boundedness of functional

In the setting of $2\pi$-periodic $C^1$ functions (whose Fourier series converge to themselves), and given a linear functional $D:C^1_{\text{per}}\to\mathbb R$ satisfying ...
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1answer
76 views

When does a left Haar measure on a locally compact group restrict to a left Haar measure on a locally compact subgroup?

Given a locally compact group which is not $\sigma$-compact, there exists a $\sigma$-compact subgroup $H$ of $G$ which is open and closed. A Remark in Folland's, A Course in Abstract Harmonic ...
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Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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65 views

definition of weak*-weak* continuous

I'm reading a paper and I have couple of terms which I can't seem to find the definition for, the first one 1) what do we mean by weak*-weak* continuous map. 2) what is the definition of a left ...
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125 views

Calderón-Zygmund operators with positive kernel

Let $T$ be a Calderón-Zygmund operator. That is, $T$ maps $L^2(\mathbb{R}^d)$ to itself and satisfies the representation formula $$ Tf(x) = \int_{\mathbb{R}^d}K(x,y)f(y)\, dy $$ for all $f \in L^2$ ...
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Definition of Left Translation of a function on a topological group

In Folland's A Course in Abstract Harmonic Analysis, he defines for a function $f$ on a topological group $G$, and $y\in G$, $$L_{y}f:x\to f(y^{-1}x)\text{ and }R_{y}f:x\to f(xy)$$ He then remarks ...
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Problem of Harmonic function.

If H is a harmonic function on an unit disk; And $H=0$ on $R_1\cup R_2$, here $R_1, R_2$ are radius of $D(0,1)$. The angle between $R_1$ and $ R_2$ is $r\pi$; here $r\in (0,1]$. If $r$ is an ...
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400 views

Haar measure on the groups SO(n) and SO(n,m)

Would you please give me some information about Haar measure on special orthogonal group SO(n) and indefinite special orthogonal group SO(n,m)? Thank you so much!
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200 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
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40 views

Show that $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$

Elias M. Stein said that by an application of Green's theorem the following equality holds $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$ where $\Delta _{S}$ is a ...
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Does $f(0) = +\infty$ when $\hat f \geq 0$ and $\int \hat f (s) \ ds = +\infty$?

Throughout, $f \in L^1(\mathbb{R})$ and $\hat f \in C_0(\mathbb{R})$ is its Fourier transform $s \mapsto \int e^{its} f(t) \ dt$. Motivation: If $\hat f \in L^1(\mathbb{R})$ too, then, by Fourier ...
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If $f \in L^1(\mathbb{R})$ and $\hat f \geq 0$, is $f$ continuous?

Suppose $f \in L^1(\mathbb{R})$. I am wondering what conditions on $\hat f = \left[ s \mapsto \int e^{its} f(t) \ dt \right] \in C_0(\mathbb{R})$ suffice to make $f$ continuous (or, more accurately, ...
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Relation between Schwartz space and Sobolev space $H_{1}$

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |x^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ and $S'(\mathbb R) ...
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93 views

Solving roots of a sum of sinusoids

Suppose I have a sinusoid with fundamental frequency $f_0$ and $N$ harmonics (all with distinct amplitudes $a_k$. Each harmonic also has it's own corresponding phase $\phi_k$ and offset $c_k$. $y(t) ...
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Is there a formula for the Haar measure on a product of groups?

Let $ (G_{n})_{n \in \mathbb{N}} $ be a sequence of locally compact topological groups with a corresponding sequence $ (\mu_{n})_{n \in \mathbb{N}} $ of Haar measures. Is there a way to construct a ...
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201 views

Why isn't the parallel between the Fourier transform and the Laplace transform complete?

I mean the question in the following sense. For Fourier, we can do it on compact intervals and then we get a sequence of coefficients. We can do it continuum-style, and then we get a superposition ...
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462 views

What does the symbol $\subset\subset$ mean? [duplicate]

In some texts (mainly complex analysis or harmonic analysis) I sometimes see the following double subset symbol $\subset\subset$ for inclusion relation of two regions, e.g., $\Omega$ and $\Omega'$ are ...
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49 views

Mean Value Property for Continuous Complex Functions

Suppose I have an open set $U$ in the complex plane and a function $g$ that is continuous on $U$. Let $C(z_0$$,r)$ be a circle fully contained in $U$ of radius $r$ whose center is $z_0$. I know ...
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42 views

Harmonic and Continuous everywhere but on a curve is harmonic throughout?

Suppose u is a harmonic function everywhere in a domain $\Omega$, but on a curve inside $\Omega$ , say a segment, and is continuous throughout, i.e $u\in C(\Omega)$. Can we conclude that u is harmonic ...
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Convolution of $L^1(G)$ functions with elements of $M(G)$.

Let $G$ be a non-discrete locally compact group with left Haar measure $\mu$. There is an isometric embedding of $L^{1}(G)\to M(G), f\mapsto fd\mu$. Since $G$ is not discrete, the point-mass measure ...
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Volume of a ball for SO(n).

Let us equip the special orthogonal group $SO(n)$ with a normalized Haar measure $\theta_n$ and let $G_r$ be the subset of rotations $\Omega$ which differ from the identity by (sufficiently small) $r$ ...
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Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
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Need help with Placherel's Theorem

We know that the law of conservation of energy dictates that the energy carried by a waveform in the time domain must equal the energy contained in its power spectrum in the frequency domain. How ...
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Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
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The spectrum of an element of the convolution algebra of a nonabelian group

Let $G$ be a locally compact group and $L^1(G)$ its convolution algebra. If $G$ is Abelian, then the spectrum of an element $f \in L^1(G)$ is equal to the image of $\hat{f}$, the Fourier transform of ...
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173 views

Help on understanding Schwartz space

Can someone give an example of Schwartz space function that doesn't decay exponentially?
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78 views

Confustion with Fourier Transform of harmonic functions

Note: this is homework. Say we assume a time-varying momentum is of the form $p(t) = \Re \{p(w) e^{-iwt} \}$ Now we would like to know the solution of the following equation, in frequency space: $ ...
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right multiplication by elements of a discrete subgroup preserve left haar measure?

If $\Gamma$ is a discrete subgroup of a locally compact topological group, G, it is not necessarily the case that right multiplication on $G$ by elements of $\Gamma$ will preserve a left Haar ...
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Matrix-valued functions with lacunary Fourier series

This question is motivated by investigation of the operator space structure of Hankel matrices (which is surely well-known to the experts). Consider a lacunary Hankel matrix, i.e. a matrix ...
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73 views

How many terms are there in a truncated Fourier series of order $N$ for a function $f: \mathbb R^n \to R$

Let $S_N(f)$ be the truncated Fourier series of order $N$ for $$ f: \mathbb R ^n \to \mathbb R. $$ How many Fourier coefficient does $S_N(f)$ contains? I'm not clear on how multidimensional Fourier ...
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64 views

Why p>1, $L^p$ and $H^p$ are essentially the same?

the conclusion comes from http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183538894 (the first page) $L^{p}$ is Lebesgue integral function space, ...
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100 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
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50 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
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412 views

Open subgroup of $SO(3)$

Does $SO(3)$ have an open nontrivial subgroup?(Group $SO(3)$ with usual matrices product, is all $3\times 3$ matrices whose determinant is 1 and for every element $A\in SO(3)$ we have $A^tA=AA^t=I_3$ ...