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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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 ...
7
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3answers
335 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in ...
1
vote
1answer
151 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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1answer
37 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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1answer
38 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 ...
12
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1answer
384 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 ...
2
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0answers
39 views

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) $r$ ...
3
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0answers
53 views

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 ...
0
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0answers
50 views

Admissibility condition for n-dimensional wavelet

Theorem 14.2.1 [S. T. Ali, J.- P. Antoine, J.- P. Gazeau, Coheret States, Wavelets and Their Generalization] The operator family $U:SIM(n)=\mathbb{R}^n \rtimes(\mathbb{R}_*^+ \times SO(n)) ...
2
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0answers
54 views

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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0answers
32 views

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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0answers
35 views

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 ...
2
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1answer
112 views

Help on understanding Schwartz space

Can someone give an example of Schwartz space function that doesn't decay exponentially?
0
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1answer
51 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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0answers
43 views

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 ...
3
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0answers
35 views

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 ...
1
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0answers
51 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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0answers
34 views

Are there orthogonal functions with continuous parameters for multivariate functions harmonic analysis?

So I was working with transforming a two-variable function $f(\theta,\phi)$ into an expansion of spherical harmonics $Y_l^m (\theta,\phi)$ such that: \begin{equation} f(\theta,\phi) = \sum_{l=0}^{L} ...
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1answer
56 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, ...
2
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0answers
94 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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0answers
60 views

Locally Compact Hypergroups

I am reading a Paper by ROBERT I. JEWETT Spaces with an Abstract Convolution of Measures. On page 51, he defines $K = \{ a,b,c\}$. The convolution operations are defined as \begin{align*} p_ap_a ...
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0answers
41 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 ...
5
votes
1answer
382 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 $A^tA=AA^t=I_3$ ...
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1answer
47 views

Existance of Haar Measure

While showing the existence of Haar measure We consider all finite sequences of positive numbers $(c_i)_{i < n}$ and all finite sets $\{x_i\mid i < n\}⊂G$ Such that $f(x)\le \sum_i c_i ...
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1answer
33 views

Existence and uniqueness of solution for an eliptic problem.

Let $\Omega \subset R^n$ be an open and bounded set with smooth boundary and $f \in C^2(\Omega)\cap C(\overline{\Omega})$. Let $ a \geq 0$ be a constant. Consider the following problem: $$ ...
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1answer
35 views

distance function is p- superharmonic?

In this article in page 3: http://arxiv.org/pdf/0904.1332.pdf says: If I consider $\Omega$ a open bounded and convex set, then the function $\delta (x) = \displaystyle\min_{y \in \partial \Omega } ...
0
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2answers
84 views

Prove that $\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |)) \in MPSH(\Omega)$

This's an example: For $u(z_1,z_1,\ldots,z_n)=\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |))$, where $z=(z_1,z_1,\ldots,z_n) \in \Omega=\mathbb{C}^n \setminus\{0\} ...
2
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0answers
38 views

Exercise 1.3.3(c) of GTM249(Classical Fourier Analysis)

Exercise 1.3.3(c) Let $0<p_0<p<p_1<\infty$ and let $T$ be an operator as in Theorem 1.3.2($\|T(f)\|_{L^{p_0,\infty}(Y)}\leq A_0\|f\|_{L^{p_0}(X)}$ for all $f\in L^{p_0}(X)$ and ...
4
votes
1answer
51 views

schwarz class and $L^2(\mathbb{R})$

Schwarz and $L^2$ both have the property that the Fourier transform is defined and bijective as a self-map of these spaces. Are they related in anyway or is this coincidence? (i.e. dual in some ...
2
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0answers
40 views

A question about BMO

In $\mathbb{R}^n$, suppose $f \in \mathrm{BMO}$ and $\phi \in \mathrm{C}_0^\infty$, then can we show that the convolution $f * \phi \in \mathrm{BMO}$?
6
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1answer
106 views

Tate's Thesis: Meaning of Local Functional Equation

I am studying the development of Tate's Thesis in Lang's Algebraic Number Theory and have a conceptual question. The setting: Let $k=\mathbb{Q}_p$. Let $\mu$ be the unique Haar measure giving ...
101
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5answers
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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 ...
8
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1answer
222 views

Wiener's theorem in $\mathbb{R}^n$

Reading Stein's "Singular integrals and differentiability properties of functions" I came across the following statement (this is in the proof of Lemma 3.2, pages 133-134): We now invoke the ...
2
votes
0answers
38 views

sequence of p-harmonic functions

Consider $\Omega$ a bounded open set in $R^n$ with $ \partial \Omega$ smooth and $u_n \in C^{1,\alpha}(\Omega)$ a bounded sequence in $C^{1,\alpha}(\Omega)$. Suppose that each $u_n$ is p - harmonic ...
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76 views

Definition of $L^p(\mathbb T)$ with $\mathbb T$ the unit circle

I'm trying to define the $L^p$ spaces in the unit circle(denoted as $\mathbb T$), as Rudin's Real and Complex analysis does in page 88. I've defined a measure in $\mathbb T$ via Riesz's representation ...
3
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0answers
52 views

Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
3
votes
0answers
137 views

Fourier coefficient of $f(x)=\frac{d}{dx}(x^{\nu} \cos(\frac{1}{x})$ for $\nu \in (0,\frac{1}{2})$.

In Zygmund's Trigonometric Series, vol I, on page 19 section 2.22 they write that Riemann showed that the Fourier coeff of $f(x)=\frac{d}{dx}(x^{\nu} \cos(\frac{1}{x}))$ for $\nu \in (0,\frac{1}{2})$ ...
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0answers
63 views

Function supported on [-1,1] with arbitrary prescribed sub-exponential Fourier decay?

Given $g:[1,\infty) \rightarrow (0,\infty)$ with $g(t) = o(t)$, does there exist $f:\mathbb{R} \rightarrow [0,\infty)$ with support contained in $[-1,1]$ such that $$ \widehat{f}(y) = ...
1
vote
1answer
55 views

To partition the unity by translating a single function

I am trying to show that there exists a (real or complex-valued) function $\psi \in C^\infty(\mathbb{R}^n)$ having the following properties: The support of $\psi$ is contained in the unit ball ...
1
vote
1answer
117 views

Subharmonic, Plurisubharmonic

Can you give me two examples of Subharmonic, Plurisubharmonic? (and not Subharmonic, not Plurisubharmonic) . Then prove that your examples. I'm looking forward to your help. Thanks.
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0answers
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Haar Measure on U(n) and O(n) [duplicate]

Every locally compact group has left-invariants haar measures. In particular, the compact groups O(n) and U(n) have them. I'm referencing the Orthogonal and unitary groups of matrix. I was wondering ...
5
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1answer
63 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 $L^p$ ...
3
votes
1answer
122 views

Why are square functions important in analysis?

I have been reading through chapter 1 of E.M. Stein's textbook Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. In chapter 1, Stein discusses the relationship ...
3
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1answer
587 views

Spherical harmonics give all the irreducible representations of $SO(3)$?

It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? ...
0
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1answer
59 views

Harmonic function 1

I have had trouble when I try to prove the following: For $u: \Omega \to \mathbb{R}$ be a harmonic function iff $u \in C^2(\Omega)$ and $\Delta u=\dfrac{\partial ^2u}{\partial ...
2
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1answer
57 views

Book searching in Pluripotential theory

Can anyone recommend me a book on pluripotential theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why ...
3
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1answer
92 views

Research paper in harmonic analysis that can be read in parallel to studying the subject.

In my idle hours I started to learn some math I touched only superficially in academia in former times. Among others I am working through the books of A. Deitmar on harmonic analysis. I've almost ...
12
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2answers
299 views

Are Fourier Analysis and Harmonic Analysis the same subject?

Are Fourier Analysis and Harmonic Analysis the same subject? I believe that they are not the same. Maybe there is big difference between those subjects but I need to know what is the main difference ...
1
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1answer
154 views

Another aspect of Heisenberg uncertainty principle

In fourier transformation theory, we have the Heisenberg uncertainty principle, i.e. Suppose $\phi$ is a function in [Schwarz space][1] which satisfies the normalizing condition ...
2
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1answer
247 views

Integrating $\sin(n\theta(x))/\sin(\theta(x))$ for some function $\theta(x)$

I have an indefinite integral of the form: $$ \int \frac{\sin(n\theta(x)))}{\sin(\theta(x))} dx. $$ $\theta$ is a function of $x$ (and actually a complicated one). Is it possible to integrate it ...