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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Convention in Riesz representation theorem vs. tempered distribution theory

We are working over the complex field here. Sometimes analysis textbooks say that every continuous linear functional on $L^p$ is integration against some $f \in L^{p'}$ for $p\in (1, \infty)$, rather ...
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Sobolev spaces vs. Hardy Spaces

I have seen sobolev spaces (the ones with the p norms of the derivatives of a multivariable function) and Hardy spaces (the objects investigated in harmonic analysis when one asks about tangential and ...
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Poisson Integral of a Lipschitz continuous function

I am reading a paper that makes reference to the following fact: Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous of some positive order $\alpha$. Let $H(x,y)$ be the extension of ...
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Does there exists $f\in A(\mathbb T)$ such that $||f||=r$ and $||\mathrm{e}^{if}||= \mathrm{e}^{r}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t)\, \mathrm{e}^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb ...
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How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
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construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
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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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How to factorize exponential as a convolution of finite number of functions( series)?

We choose a points $x_{1},..., x_{n}$ in $\mathbb Z$ (not 0), such that $$x_{k+1}\not = x_{1}+x_{2}+...+x_{k} \ \ (k=1,2,...,n-1).$$ Let $\delta_{x}$ denote the measure of total mass $1$, ...
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How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
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Show that two series are equal

In my harmonic analysis class I have to prove that for all $a>0$ the following equality holds: $$\sum_{n\in \mathbb{Z}}e^{-n^2\pi a}=\frac{1}{\sqrt{a}}\sum_{k\in \mathbb{Z}}e^{-k^2 \pi / a}.$$ I'd ...
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Is the p-adic Schwartz function uniform continuous?

In p-adic case, Schwart function is the function which has compact support and locally constant. But can we say its uniform continuity from this? I think it would not be true, but I am not certain ...
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How to inter change of norm and limit in the Banach algebra?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
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integrability of Hilbert transform of a function [duplicate]

The problem is : Let $\varphi \in \mathcal S(\mathbb R)$ (Schwartz space) with $\int \varphi \ dx = 0$. Then the Hilbert transform of $\varphi$ belongs to $L^1(\mathbb R)$. I believe this helps ...
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Problem in harmonic analysis

suppose $p$ be a fixed psitive real number and $f$ is an entire function with $$\lvert f(0) \rvert^p=\int_\mathbb{C}\ \lvert f(z)\exp(-\alpha\lvert z \rvert ^2) \rvert^p dA(z) $$ where $\alpha ...
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about the classical Hardy Littlewood Sobolev inequality

The Hardy - Littlewood - Sobolev inequality says : Let $0< \alpha < N$ , $1 \leq p,q < \infty$ with $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{N}$, Consider $I_{\alpha} f (x) = c_{\alpha , ...
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The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
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Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
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Does there exists $f\in L^{1}(\mathbb R)\cap FL^{1}(\mathbb R)$(=Fourier algebra) but $|f|\not \in A(\mathbb R)$?

For $f\in L^{1}(\mathbb R)$; We define the Fourier transform of $f$ as follows: $$\hat{f}(\xi)= \int_{\mathbb R}f(x) e^{-2\pi i \xi x}dx; \ \text {for} \ \xi \in \mathbb R.$$ Consider a Fourier ...
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Can one visualise the dual groups to Cantor groups?

My question is very simple, and, probably an answer can be found in any harmonic analysis textbook, but it seems I have failed that task. It occurred to me that I don't understand the structure of the ...
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Interesting equation in L^1

Consider $L^{1}(T) = \{ f : R \rightarrow C \text{ with period 1 and } \int_{0}^{1} |f (x)| \ dx < \infty\}$. For $f,g \in L^{1}(T)$ the convolution is given by $(f * g)(x)= ...
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Basics of Haar measure

Suppose $G$ is a locally compact group. Then $G$ has a left-invariant measure $dg$, say, which means that $$\int f (hg) dg = \int f(g) fg$$ for any test function integrable on $G$. The ...
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Doubt in the definition of the Fourier transform in $L^{2}(\mathbb R^n)$

I am trying to understand the definition of the Fourier transform in $L^{2}(\mathbb R^n)$ . I am understand of this manner : Let $f \in L^{2}(\mathbb R^n)$ and $n$ a natural number. Define $f_n = f ...
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Fitting Sounds Waves with Sines/Cosines

I am trying to model sound waves with a series of sines and cosines but I am not sure what the best way to find the best deterministic sine/cosine combination that best fits the data. What are some ...
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Boundary of real part of functions in $H^p$ and Poisson nontangential maximal function

I have two questions when reading on $H^p$ spaces, many books do not give their proofs. First we reminde that $H^p(\mathbb R^2_+)$ consists of all functions $F$ which is analytic in the upper half ...
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Abstract Fourier Analysis

I am trying to prove that given a locally compact abelian (Hausdorf) topological group, the characters on it parameterize the multiplicative linear functionals on the banach * algebra $L^1(G, d\mu)$ ...
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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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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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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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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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1answer
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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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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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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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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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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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The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
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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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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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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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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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If a normal subgroup, N, contains a lattice why does G/N have finite measure?

Suppose $G$ is a locally compact Hausdorff topological group and suppose $H \leq N \leq G$ are closed subgroups with $N$ normal. Now suppose $G/H$ has a finite $G$-invariant Boreal measure (in the ...
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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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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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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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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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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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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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1answer
31 views

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 ...