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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32
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849 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
9
votes
1answer
1k views

Theorem of Steinhaus

The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also ...
17
votes
3answers
573 views

A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
14
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1answer
668 views

Properties of Haar measure

Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
14
votes
3answers
960 views

Do discontinuous harmonic functions exist?

A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, ...
13
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1answer
6k views

Criteria for swapping integration and summation order

I have a function (a potential from an electrostatic potential via a Fourier series) in the form of $$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv$$ ...
3
votes
1answer
321 views

Asymptotic error of Fourier series partial sum of sawtooth function

In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement: $$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$ where $\{x\}$ denotes the ...
9
votes
1answer
689 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
3
votes
1answer
818 views

Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
9
votes
2answers
407 views

Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
7
votes
1answer
693 views

Reference request: Fourier and Fourier-Stieltjes algebras

I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...
9
votes
1answer
740 views

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 $\alpha$. ...
6
votes
3answers
625 views

Derivatives distribution

Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that $$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$ Then how to prove that $f$ is a constant? I had ...
3
votes
0answers
95 views

convolution of $f$ and $g$ is in $L^p$ where $f$ has compact support.

Suppose 1≤p≤∞ and $f∈L^1(G)$ and $g∈L^p(G)$. I know this is true $‖f*g‖_p ≤‖f‖_1 ‖g‖_q$ , Where ∗ is convolution of f and g. I want to prove that If G is not unimodular, we still have g∗f is in LP ...
2
votes
1answer
78 views

A simple question of Littlewood-Paley decomposition.

Let $\{f_k(x)\}_{k=0}^\infty$ be a Littlewood-Paley decompositon, that is, $$ f_k \in C_c^\infty $$ $$ \sum_{k=0}^\infty f_k (x) = 1,$$ $$ \text{supp} f_0 \subset \{ |x| \leq 2 \},$$ $$ \exists f ...
1
vote
1answer
130 views

Suppose G is unimodular. If f is in LP(G) and g is in Lq(G) where 1 < p,q < ∞ and 1/p+1/q=1, then f * g ∈Co(G) and ||f * g||sup ≤ ||f||p ||g||q

Suppose G is unimodular. If f is in LP(G) and g is in Lq(G) where 1 < p,q < ∞ and 1/p+1/q=1, then f * g ∈Co(G) and ||f * g||sup ≤ Ilfllp llgllq. * is convolution f and g. I read the ...
0
votes
1answer
185 views

How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...
108
votes
5answers
7k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
30
votes
2answers
547 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
5
votes
1answer
120 views

Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
7
votes
2answers
380 views

Good book on topological group theory?

I'm looking for a good introduction to the theory of locally compact groups and their representations. It may assume the reader to be familiar with basic group theory, topology and measure theory.
16
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1answer
248 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 ...
12
votes
1answer
512 views

Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence ...
12
votes
1answer
564 views

What are the differences and relations of Haar integrals, Lebesgue integrals, Riemann integrals?

Are Riemann integrals special cases of Haar integrals? Why do we need the invariant property under some actions of groups in the definition of Haar integrals? For example, if we have a group of real ...
3
votes
2answers
861 views

Why are translation invariant operators on $L^2$ multiplier operators

For $m \in L^\infty$, we can define the multiplier operator $T_m \in L(L^2,L^2)$ implicitly by $\mathcal F (T_m f)(\xi) = m(\xi) \cdot (\mathcal F T_m)(\xi)$ where $\mathcal F$ is the Fourier ...
2
votes
2answers
154 views

An integral$\frac{1}{2\pi}\int_0^{2\pi}\log|\exp(i\theta)-a|\text{d}\theta=0$ which I can calculate but can't understand it.

I have encountered this integral:$$\frac{1}{2\pi}\int_0^{2\pi}\log|\exp(i\theta)-a|\text{d}\theta=0$$ where $|a|<1$ in proving Jensen's formula. because I am stupid, I don't know why it is equal ...
12
votes
3answers
506 views

Rate of divergence for the series $\sum |\sin(n\theta) / n|$

In the following we consider the series $$ S(N;\theta)= \sum_{n = 1}^{N} \left| \frac{\sin n\theta}{n} \right| $$ parametrized by $\theta$. It is well known that this series (taking the limit ...
6
votes
1answer
350 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 $n\in (0,d]$ ...
5
votes
2answers
199 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 ...
5
votes
1answer
336 views

Harmonic functions on $\mathbf{Z}^2$

Problem 1: Find all functions $f:\mathbf{Z}^2 \to \mathbf{R}$ which are harmonic in the sense that $$f(x,y) = \frac{f(x+1,y) + f(x-1,y) + f(x,y+1) + f(x,y-1)}{4}$$for all $(x,y)\in\mathbf{Z}^2$, ...
3
votes
1answer
483 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 ...
3
votes
1answer
312 views

Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary ...
2
votes
0answers
37 views

Proving $\mu\ast K_n\to\mu$

Let $\{K_n\}$ be approximating unit and $\mu\in M(\mathbb{T})$. Show that $\mu\ast K_n\to \mu$ weakly means $$\int f(t)d(\mu\ast K_n)(t)\to\int f(t)d\mu(t)$$ Suppose $\mu\in L^1(\mathbb{T})$, I ...
4
votes
1answer
506 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 ...
2
votes
1answer
87 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 ...
2
votes
2answers
782 views

Bounded linear operators that commute with translation

I'm trying to read Elias Stein's "Singular Integrals" book, and in the beginning of the second chapter, he states two results classifying bounded linear operators that commute (on $L^1$ and $L^2$ ...
1
vote
0answers
31 views

Existence of Inverse Laplace Transform

Let $F(s)$ the Laplace transform of a function $f(t)$ . Under which conditions on $f(t)$ there exist a unique $g(t)$ such that $g(t) = \mathcal{L}^{-1}\{e^{- F(s)}\}(t) \quad $ ? ...
1
vote
1answer
280 views

Transitive group actions and homogeneous spaces

Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is ...
0
votes
1answer
84 views

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.
10
votes
1answer
227 views

Suppose $\phi$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$. Then $\phi\in W^{2,1}$

I'm trying to prove the statement in the title in as simple a way as possible. It is Theorem 3.2.9 in Helein's book "Harmonic maps, conservation laws, and moving frames", although it is not proved ...
6
votes
4answers
2k 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. ...
5
votes
1answer
88 views

Function invariant under Hilbert transform

Let $f\in C_0^\infty(\mathbb{R}^n)$ and $$Hf(x)=p.v.\int_{\mathbb{R}}\frac{f(x-y)}{y} dx$$ the Hilbert transform of $f$. Is it possible that $Hf=f$ (a.e. and possibly after extending $H$ to $L^p$ ...
5
votes
1answer
200 views

A Schwartz function problem

Let f be a strictly positive Schwartz function on $\mathbb R$. Does it imply $\sqrt f$ is a Schwartz function on $\mathbb R$?
4
votes
1answer
181 views

Why are Haar measures finite on compact sets?

I'm working through the answer by t.b. to another user's question here: A net version of dominated convergence? because I am trying to work through a related problem and I think it will be ...
4
votes
1answer
113 views

number of zeros of the superposition/interference of sine oscillations

There is a tricky problem to solve and we ask for your kind help. In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases ...
3
votes
1answer
29 views

Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
3
votes
2answers
128 views

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family ...
3
votes
1answer
70 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 ...
3
votes
2answers
290 views

Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
3
votes
1answer
219 views

Surjective endomorphism preserves Haar measure

How to prove the following statement: Let $G$ be a compact topological group and let $m$ be the Haar measure on it. Let $\varphi$ be a continuous endomorphism of $G$ onto $G$, i.e., the map $\varphi$ ...