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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11
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1answer
374 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
3
votes
1answer
286 views

Harmonic analysis in number theory

When I was reading Folland's A course in abstract harmonic analysis, I was told these materials have wonderful applications to number theory. However, I do not see really a lot of examples there. Can ...
4
votes
2answers
177 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
1
vote
0answers
21 views

A question about the Littlewood-Paley decomposition. [duplicate]

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 ...
2
votes
1answer
81 views

A property of Littlewood-Paley decomposition

Let $f_j \in C_c^\infty$ and assume there exists $M>0$ such that $\text{supp}f_0 \subset \{ |x| \le M \}$ and $\text{supp} f \subset \{ 1/M \le |x| \le M \}$. Define $f_j$ by $$ f_j (y) = f(y/2^j) ...
2
votes
1answer
81 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 ...
-3
votes
1answer
55 views

How to prove $\mathrm{supp} ~ R(t)\in \{|x|<t\}$

Let $R(t)$ define as $$R(t):=F^{-1} \left(\frac{\sin |\xi|t}{|\xi|} \right),$$ how to show that $$\mathrm{supp} ~ R(t)\in \{|x|<t\},$$ where $F$ is the Fourier transform.
0
votes
0answers
107 views

Zero-crossings of periodic function

Given a periodic function of the form $$ s(\theta) = \sum_{h=1}^H a_h \cdot \cos(h\cdot \theta + \phi_h),$$ how can I determine its zero-crossings $\theta_0$?
2
votes
0answers
118 views

expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
3
votes
1answer
143 views

Uncertainty principle in harmonic analysis

Given a function $f$ (in some suitable (EDIT: nice) function space) supported on a ball $B\subset\mathbb{R}^n$ of radius $R>0$, I have commonly heard people, who understand these things, say stuff ...
6
votes
2answers
202 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 ...
1
vote
1answer
92 views

Question which involves Riesz potential

Let $f :S^{n-1}\longrightarrow R_+$ , $P_{j,k}. j,k \in N$ be a spherical harmonics (http://en.wikipedia.org/wiki/Spherical_harmonics) and $\displaystyle{f\left(\frac{x}{|x|}\right)|x|^{1-\frac ...
3
votes
1answer
154 views

Corollary from Khintchine's inequality

Let $z_1,\dots,z_n\in\mathbb{C}$ and $\epsilon_j\in\{-1,1\}$ for $j=1,\dots,n$ independent random variables with $P(\epsilon_j=\pm 1)=1/2$. Khintchine's inequality states that ...
4
votes
1answer
1k views

Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
2
votes
1answer
84 views

Application of Green's theorem to probability

I encountered this problem while reading a statistic text. Since I am not quite familar with the background knowledge. Wonder can someone help me to explain the details of the following proof? ...
0
votes
1answer
169 views

Harmonic mean of absolute value squared discrete Fourier transform

Let $X[k] \in \mathbb{C}$ be the discrete Fourier transform of $x[n] \in \mathbb{R}$, where $k,n = 0,1,2,...,N-1$. Parseval's theorem relates the arithmetic mean (AM) of absolute value squared ...
0
votes
1answer
67 views

Hilbert transform and maximal function

What is the relation between the Hilbert transform (in $\mathbb{R}$) and maximal functions?
4
votes
1answer
133 views

Use of the Littlewood-Paley decomposition to recover the $H^s$ norm

Let $\phi\in C^{\infty}_0(\mathbb{R}^n)$ be such that $$\{\lvert \xi\rvert \le 1\} \prec \phi \prec \{\lvert \xi \rvert < 2\}^{[1]} $$ and define the Littlewood-Paley projectors as ...
5
votes
1answer
93 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

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
0
votes
0answers
56 views

approximate Fourier transform

Let $\mathcal{F}$ stand for the Fourier transform. Suppose $f : [-\delta/2,\delta/2] \to \mathbb{C}$ is a "nice" function. Is it true that $$\left|\mathcal{F} \left(e^{imx} \left(e^{ix^2}-1 ...
3
votes
2answers
86 views

Estimating time in harmonic signal

I hope someone can help me with the following problem: Assume a periodic signal of the form $$\begin{align} s(t) &= \sum\limits_{p=1}^P \sin(p\Omega_0t)\\ &= \sum\limits_{p=1}^P ...
1
vote
1answer
57 views

Uniform Rectifiability

What is the definition of uniform rectifiability as used in the context of analytic capacity of compact sets in $\mathbb{C}$? The precise context is this paper by Mattila, Mernikov and Verdera.
7
votes
2answers
237 views

Interpolation using trigonometric polynomials of bounded modulus

Consider a grid of points $T=\{t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to derive conditions on $t_1,\ldots,t_m$ (interpolation points) under which for any sequence of complex numbers ...
5
votes
1answer
177 views

Where should the 2$\pi$ go in the Fourier Transform?

In some lecture notes on Harmonic Analysis from Terence Tao here, he defines the fourier transform by $$\hat{f}(\tau)=\int_{\mathbb{R}}e^{-2\pi i t\tau}f(t)dt$$ and then says This is really the ...
0
votes
1answer
131 views

A question of positively homogeneous functions

Let $f$ be a positively homogeneous function of degree $k$, i.e., $$ f(x, \lambda y) = \lambda^k f(x, y)$$ for any $(x,y) \in \mathbb R^n \times \mathbb R^n, \lambda >0$. Then how can I show that $ ...
2
votes
0answers
83 views

Proof of Caratheodory's theorem about the unique determination of a linear combination of sinusoids

Following is a statement of Caratheodory's Theorem about a positivelinear combination of sinusoids :- Any positive linear combination of k sinusoids is uniquely determined by its value at time t ...
2
votes
0answers
121 views

Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm

If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
2
votes
1answer
64 views

Proof that Muckenhoupt's $A_q$ Condition Implies $A_p$ for $p<q$?

It is said $f\in A_p$ if it satisfies the following (Muckenhoupt's $A_p$) condition: ...
4
votes
1answer
336 views

Proof of weak maximum principle.

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
4
votes
1answer
507 views

Example of an unbounded operator

Can somebody give me an easy example of a linear operator which maps $L^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$ and $L^\infty(\mathbb{R}^n)$ to $L^\infty(\mathbb{R}^n)$ (but not boundedly) but does ...
2
votes
1answer
118 views

Carleson embedding theorem

The Carleson embedding theorem gives a criterium by which to decide when for a measure $\mu$ the operator that takes a function on the real line to its harmonic extension (by convolution with Poisson ...
2
votes
1answer
176 views

Proof that $dx/|x|$ is a Haar measure on non-zero reals?

Most importantly, what is the meaning of this notation $\lambda = dx/|x|$? How do I compute say $\lambda(0,1)$ for example?
3
votes
1answer
121 views

Weak $L^{p}$ spaces are quasi-normed?

Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition $L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that ...
2
votes
1answer
131 views

Showing atomic $H^{1,p}$ is a Banach Space

Consider $1<p\leq\infty$ and let $H^{1,p}=H^{1,p}(\mathbb{R}^n)$ be an atomic Hardy space, that is $H^{1,p}\subset L^1(\mathbb{R}^n)$, and $f\in H^{1,p}$ if there exists a sequence of $p$-atoms ...
7
votes
2answers
415 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.
2
votes
1answer
210 views

Fourier analysis questions

Can anyone give me a hand with the proof of this properties? Prove that: a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
3
votes
1answer
70 views

Are the continuous functions on $G$ dense in $L^{1}(G)$?

If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
4
votes
1answer
196 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 ...
3
votes
0answers
87 views

Are Haar measures complete?

If $G$ is a locally compact group and $\mu$ is a left Haar measure for $G$, then is the measure space $(G,B(G),\mu)$ complete (where $B(G)$ is the set of Borel subsets of $G$)? Or do we have to take ...
1
vote
1answer
118 views

Why is logarithm in BMO

I read that $\log|x|$ is supposedly a typical example of a BMO function. How do I see that it is in BMO actually?
1
vote
0answers
22 views

Convergence of an Integral in a locally compact group

I'm trying to finish an exercise which I asked about earlier here: Mapping $G$ into its group algebra as left multiplication. Continuous? $\bf{\text{The setting:}}$ Let $G$ be a locally compact ...
0
votes
0answers
32 views

Bounding the partial derivatives of $\left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}$.

Let $\tau = (\tau_1,\ldots,\tau_n) \in \mathbf{R}^n$, $s\geq 0$, and define the function $f : \mathbf{R}^n \to \mathbf{R}$ by $$ f(\xi) = \left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}. $$ ...
4
votes
1answer
110 views

Involution in $L^{1}(G)$ is isometric.

(Sorry for asking so many questions of the same type. There is an underlying issue that I think once resolved will allow me to understand them all at once.) Let $G$ be a locally compact group, and ...
2
votes
1answer
137 views

How to use the modular function of a locally compact group?

Let $G$ be a locally compact group with left Haar measure $\mu$. Let $\delta:G\to(0,\infty)$ the unique modular function such that $\delta(x)\mu(E) = \mu(Ex)$ for all Borel sets $E$ and $x\in G$. ...
1
vote
3answers
354 views

Introduction to Abstract Harmonic Analysis, reference suggestions?

I'm looking for a good starting book on the subject which only assumes standard undergraduate background. In particular, I need to gain some confidence working with properties of Haar measures, so I ...
3
votes
0answers
88 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
2
votes
2answers
159 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 ...
3
votes
3answers
602 views

Fourier transform of a compactly supported function

In which space does the Fourier transform of a smooth compactly supported function $\phi$ lie? I would not say it lies in $\mathcal{S}$, heuristically as one can approximate the step function which is ...
1
vote
1answer
44 views

Characterisation of the spectrum of certain unitary representations on $L^2(G)$

I have been informed of the existence of theorems in harmonic analysis that will allow me to calculate the spectrum of given unitary operators $L^2(G)$ where $G$ is a locally compact group. So far I ...