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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Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
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How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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“Dual Lorentz characterization of L^q”

I am trying to do Exercise A.4 of Terence Tao's book on nonlinear dispersive equations (available here: http://www.math.ucla.edu/~tao/preprints/chapter.pdf, the exercise is on p.342 in Appendix A). ...
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72 views

Fourier transform of a a function in the space

Which is the Fourier transform (in the sense of distributions) of the function $f(x)=x/\|x\|^n $, where $x$ belongs to the Euclidean space $ R^n$?
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104 views

Laplace operator and Fourier transform

Let $f$ be a function with compact support in $D\subset \mathbf{R}^2$. Is is known that $$-\Delta f=F^{-1}|x|^2 F f,$$ where $F$ is the Fourier transform. Which is the $n-$dimensional analogous ...
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411 views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
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1answer
40 views

How to apply the method of stationary phase here?

Consider the following oscillating integral $$ I(\xi;t) = \int\limits_{\mathbb R}\int\limits_{\mathbb R^n} e^{it(\xi y - \theta f(y))}a(y) \, dy \, d\theta, \quad \xi \in \mathbb R^n \setminus 0, ...
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47 views

How to determine measure from the integral equation?

Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...
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12 views

Fast transpose of undecimated wavelets

I am using undecimated Daubechies wavelets and I need to compute the forward and adjoint of the wavelets several times. I am using the Rice Wavelet Matlab toolbox ...
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33 views

When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral ...
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47 views

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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1answer
36 views

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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1answer
42 views

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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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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60 views

An $L^1$ function whose Fourier series converges but not to itself

Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?
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17 views

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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37 views

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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1answer
32 views

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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1answer
55 views

Solving the equation $\int G(t) dt =\frac{\sin x}{x}$

I have to solve the equation $$\int_{\mathbb R} \frac{f(t)}{1+(x-t)^2} dt =\frac{\sin x}{x}.$$ I tried change of variables to make the $\frac{1}{1+(x-t)^2}$ part resemble $e^{h(x)}$ so I can use the ...
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1answer
52 views

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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1answer
23 views

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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1answer
28 views

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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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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24 views

integrability of Hilbert transform of a function

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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41 views

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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1answer
66 views

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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1answer
121 views

Are the two definitions of the complementary Young function equivalent?

The description of the problem: For a Young function I would refer the reader to the book "Function spaces" by Pick Luboš, Kufner Alois, John Oldrich and Fucík Svatopluk, and published by de Gruyter. ...
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1answer
45 views

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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51 views

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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1answer
41 views

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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35 views

Determine polynomials with $n$-variables

Here is a funny problem arise from harmonic analysis: Let $E$ be a measurable subset of $\mathbb R^n$ with $m(E)>0$, where $m$ is the usual Lebesgue measure on $\mathbb R^n$. In practice, $E$ ...
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1answer
32 views

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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47 views

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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50 views

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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1answer
101 views

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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1answer
35 views

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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2answers
50 views

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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4answers
153 views

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
54 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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1answer
44 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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1answer
43 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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2answers
75 views

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 $$ \|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}. $$ ...
3
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1answer
51 views

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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1answer
50 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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1answer
24 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
79 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 ...