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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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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0answers
24 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ has the cancellation property, has at least one full derivative, ...
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1answer
20 views

How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact. Fix $x_{0}\in \mathbb R.$ My Question is : Can we expect to choose, $\phi \in ...
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1answer
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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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0answers
55 views

Dual Lorentz characterization of $L^q$

I am trying to do Exercise A.4 of Terence Tao's book Nonlinear dispersive equations: local and global analysis. The exercise is on p.342 in Appendix A. The problem statement is: Let $f\in ...
2
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2answers
68 views

The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
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3answers
451 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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2answers
71 views

Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that ...
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0answers
33 views
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1answer
35 views

If A has positive Haar measure then $AA^{-1}$ is a neighborhood of $e$

I read the following exercise: Prove that if $G$ is a locally compact topological group with Haar measure $\mu$ and $A \subset G, \mu (A) >0$, then $AA^{-1}$ contains an open neighborhood of the ...
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1answer
70 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
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1answer
70 views

Noncommutative Fourier Transform

The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My ...
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2answers
69 views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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0answers
23 views

The variational formulation of entropy

For $f:\mathbb Z_2^n \to [0, \infty)$, the entropy of $f$ is defined as $$ {\rm Ent}(f) = \mathbb E[f(X) \log f(X)] - \mathbb E f(X) \log(\mathbb E f(X)), $$ where $X$ is a random element of $\mathbb ...
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0answers
131 views

Why is the derivative of the translates of a measure measurable?

Let G be a topological group and X a measure space. Let $G \times X \rightarrow X$ be a measurable group action, $\mu$ a $\sigma$-finite measure on $X$, and $g\mu$ (for any $g \in G$) the measure ...
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1answer
56 views

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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1answer
65 views

Locally compact groups

Let $S= G_1\bigcup G_2 $, where $G_1$ and $G_2$ are two groups. If $S$ is locally compact, is it true that either $G_1$ or $G_2$ is locally compact? Thank you.
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1answer
60 views

Amenability of finite dimensional norm algebras

Let $(\cal A,\|\cdot\|)$ be a finite dimensional norm algebra (Banach Algebra). Can we say any thing about the amenability of $\cal A$. What if we impose some extra conditions on $\cal A$, say ...
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0answers
12 views

Describing function of a non linearity with memory

Can anyone help me on finding the correct methodology to compute the describing function of the following NL function? ...
2
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1answer
63 views

Hardy Space Cancellation Condition

I have been reading the chapter on Hardy spaces in Stein's Harmonic Analysis book, and I am having a lot of trouble figuring something out. The setting here is $\mathbb{R}^n.$ Let $f \in L^q$ be ...
7
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1answer
32 views

If $w$ is in weak $A_{\infty}(d\mu)$ where $d\mu$ is a doubling measure, then is $w\,d\mu$ doubling?

Let $\mu$ be a positive Borel measure on $\mathbb{R}^n$ and let it be doubling i.e. there exists a a constant $C>1$ such that $\mu(B(x_0, 2r)) \leq C \mu(B(x_0,r))$ for all balls $B(x_0,r)$. Let ...
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4answers
761 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. ...
3
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1answer
47 views

About p-adic numbers

I'm studying the dual group of the diadic rationals, $\widehat{\mathbb{Z}[1/2]}$, where the dual is the dual of Pontryagin of $\mathbb{Z}[1/2]$. In some papers says that $\widehat{\mathbb{Z}[1/2]}$, ...
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2answers
32 views

Very simple question regarding sum/difference identity

If I have $\sin(0.7x-47t+C)$ where do I carry my constant $C$? The same with my sum-to-product identities. This problem is showing up for me because I'm studying mechanical waves at the moment. I ...
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1answer
193 views

A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the ...
7
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1answer
189 views

Using normal families to bound a complex integral

I am trying to prove that $$\int_{\partial T(Q)} |F'(z)| \,ds(z) \lesssim \int\int_{T(Q)} |F'(z)| |\varphi'(z)|^2 \log \frac{1}{|z|} \,dx\, dy$$ This is an estimate on page $6$ of this paper by ...
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0answers
26 views

Why the $L^1(R)$ space does not have a unconditional basis

Why the $L^1(R)$ space does not have a unconditional basis. It is a known fact that $L^p(R)$ ($1<p<+\infty$) has unconditional basis. A simple example is the dyadic wavelet or Haar system. I ...
3
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0answers
48 views

Building up the Fourier inversion theorem on locally compact abelian groups

I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ...
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1answer
11 views

Fourier coefficients of a measure and absolute continuity

A relative of a theorem of Peyriere (found in a 1997 paper of Klemes and Reinhold, "Rank One Transformations with Singular Spectral Type") says that if $\mu$ is a Borel probability measure on $S^1$ ...
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0answers
56 views

An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
3
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1answer
85 views

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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1answer
41 views

proof that$ L^1 (G)$ is a subspace of $M(G)$

Let G be a locally compact group, and let $M(G)$ be the space of complex Radon measures on G. I identify the function f with the measure $f(x) \rm dx$ . but How do I prove this inclusion?؟ . .
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Haar measure on $ \mathbb{R} × \mathbb{T}$ and on dual $\mathbb{R} × \mathbb{T}$

I've solved this exercise somewhat.To complete it please help me Haar measure on G $=$ translation invariant on G $$ μ(A)=μ(A+t)$$ if $ G=\mathbb{R}$ then Haar measure on G is lebesgue measure. and ...
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0answers
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Uniqueness in Bochner's theorem

Bochner's theorem : Let $G$ be a locally compact Abelian group. Then for any $ \phi \in \ P(G) $ there is a unique positive Radon measure $ \ μ \in \ $ M ($ \widehat{G} $) such that ...
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2answers
715 views

Approximation of the identity and Hardy-Littlewood maximal function

The inequality seems to be simple but I could not find the right limits of integration. $$\sup_{\delta>0} |f*K_{\delta}|(x)\leq c f^*(x)$$ Where is some positive constant, $f$ is integrable, ...
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0answers
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$\widehat{\mathbb{T}}$ can be identified with $\mathbb{Z}$

$ \mathbb{T} \stackrel{\text{def}}{=} \{ z \in \mathbb{C} : |z| = 1 \}$ $\widehat{\mathbb{T}} \stackrel{\text{def}}{=} \text{Hom}(\mathbb{T},\mathbb{T})$ To show that $\widehat{\mathbb{T}}$ can be ...
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1answer
22 views

Local Module Homomorphism

Let $A$ be Banach algebra and $X,Y$ be two left Banach $A$-modules. It is said that the linear bounded map $\phi:X\to Y$ is left $A$-module homomorphism if for any $a\in A$ and any $x\in X$ we have ...
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Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ ...
3
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1answer
67 views

The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
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1answer
67 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ ...
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1answer
30 views

When is a function a Fourier transform of an integrable function?

Specifically, in the case $f(\xi)=\frac{1}{(1+\xi ^2)^\epsilon}$ where $0<\epsilon<1$. I wish to prove this is a Fourier transform of a $L_1$ function. Any insight into the manner would be ...
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1answer
42 views

Inequality between Fourier coefficients implies inequality for $L^p$ norms on the circle

Given two functions from $L_p [-\pi,\pi]$, where $p\geq 2$, $p$ is an even integer, and $f_n>|g_n|$ for every $n$ (where $f_n$ is the $n$th Fourier coefficient), I need to prove that ...
3
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1answer
57 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
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1answer
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Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
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1answer
20 views

Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
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1answer
33 views

Locally compact group and continuous function with compact support

Suppose that $G$ is a locally comact group, let $H$ be a open subgroup of $G$ and let $\phi:G\to\Bbb C$ be a continuous function such that $Supp\phi=:\overline{\{x\in G: \phi(x)\neq 0\}}$ is compact. ...
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1answer
43 views

Open subgroup and group algebra

Let $G$ be a locally compact group and $H$ be an open subgroup of $G$. Consider the group algebras $L^1(G)$ and $L^1(H)$ with convolution product and consider $L^1(H)$ as a subalgebra of $L^1(G)$ ...
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1answer
345 views

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 ...
4
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1answer
45 views

Weak* convergence in $(L^q)^*$ and convergence in $L^p$

Let $\{\phi_n\}$ be a sequence in $L^p(X)$. Assume that $\phi_n\to \phi$ weak* under the natural identification $L^p\cong (L^q)^*$. Of course, it is not true in general that $\phi_n\to \phi$ in $L^p$ ...
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1answer
12 views

Discontinuous characters on locally compact abelian groups

I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither ...