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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130 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
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1answer
90 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 $L^p$ ...
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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 ...
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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 ...
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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
55 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.
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2answers
234 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
175 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
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1answer
125 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 $ ...
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0answers
81 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 ...
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0answers
118 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
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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
320 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
489 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
116 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
120 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
129 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 ...
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2answers
400 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
69 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
194 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 ...
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0answers
85 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 ...
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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?
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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 ...
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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
108 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
135 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 $x\in G$. ...
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3answers
334 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 ...
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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
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2answers
158 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
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3answers
585 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 ...
4
votes
1answer
114 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 ...
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1answer
98 views

Harmonic Measure & Brownian Excursion

I have a disk of radius R removed from the plane. Brownian excursions are emanated from infinity. I am tasked to find a probability that the brownian motion impacts an arc of the disk boundary. I am ...
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95 views

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and ...
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1answer
80 views

Measure of the boundary of a union of cubes.

Suppose that we are given a collection of dyadic, mutually disjoint, open cubes in $\mathbf{R}^n$ in which the the union of all the cubes has finite measure. Is it necessary that the boundary of the ...
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2answers
422 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 ...
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0answers
199 views

DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?

I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation: Say, I have a function vector with ...
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1answer
22 views

An approximation of lengths dealing with elements of cubes. Show $|x-y| \approx \ell(Q) + |x - c_Q|$.

So the problem I have at hand should be rather elementary, but I can't figure out. It's in a proof I'm trying to understand completely. Here's what I need to do exactly: Show that $|x-y| \approx ...
1
vote
1answer
44 views

Extracting Harmonic series components

I have a number which is made up of a Harmonic series. 1/2 + 1/3 + 1/4 etc. Some of the components may not be in the number.. 1/2 + 1/7 + 1/11 etc. Is it possible to recover the individual ...
2
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1answer
83 views

Why does this inequality for all characteristic functions imply it for simple functions?

This question is probably obvious, but I'm not seeing how to obtain it. A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and ...
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1answer
202 views

Showing a Schwartz Function Bound

I have a question on how to get the correct upper bound of a Schwartz function. Unfortunately, I've never understood this even though I've seen my professors do it a thousand times. I figured it's ...
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1answer
58 views

Does $|T(f) - T(g)| \leq |T(f-g)|$ hold for a sublinear operator $T$?

Let $X,Y$ be function spaces with functions taking values in $\mathbf{C}$. An operator $T:X\to Y$ is called sublinear if for all $f,g \in X$ and all $\lambda \in \mathbf{C}$, we have $$ |T(\lambda f)| ...
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1answer
69 views

Lorentz Space property: $\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$

I would like to understand a statement similar to Fatou's Lemma in the Lorentz space setting. It is as follows. Suppose $0 < q,s < \infty$ and $f_n,f$ are measurable functions on a ...
3
votes
2answers
142 views

Identity involving partial sums of Fourier series

Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$ How can I show that ...
2
votes
1answer
36 views

Given $\widehat{\varphi} = \chi_{[-1/2,1/2]}$, show the system $\{\varphi(2^v x)\}_{v \in \mathbb{Z}}$ is not orthogonal

If $$\widehat{\varphi} = \chi_{[-\frac{1}{2},\frac{1}{2}]}$$ then it is not hard to compute via the inverse Fourier transform that $$\varphi(x) = \frac{\sin(\pi x)}{\pi x}$$ so we need to show ...
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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.
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71 views

Fourier algebra and multipliers of a finite discrete group

Let $G$ be a finite discrete group. We denote by $A(G)$ the Fourier algebra of $G$ and $M_{cb}A(G)$ the space of completely bounded multipliers of $A(G)$. Is is true that $A(G)=M_{cb}A(G)$ ...
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1answer
175 views

Which spherical harmonic function will correspond to such a representation?

On Wiki there's a figure displaying "visual representations of the first few spherical harmonics." I was wondering exactly which spherical harmonic function will generate a representation like this ...