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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Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
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81 views

Harmonic analysis on discrete groups

I am currently confused about the notion of generalized Fourier transforms on discrete groups (I'm only concerned about discrete groups here, not necessarily abelian). On finite discrete groups, I ...
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Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...
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What are the “right” spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace ...
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Hilbert transform on $L^p(\mathbb{T})$

Let $\infty >p\geq 2$, then for $f\in L^p(\mathbb{T})$ (here $\mathbb{T}=[0,1)$), show that for any real-valued trigonometric polynomial $f$, we have $H(f^2-(Hf)^2)=2fHf$. The hint is to use the ...
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Looking for a proof involving the Harmonic number [duplicate]

Prove that: $\displaystyle \sum_{k=1}^{\infty} \frac{H_k}{k^q} = (1 + \frac{q}{2})\zeta(q + 1) - \frac{1}{2}\cdot \sum_{n=1}^{q-2}\zeta(k+1)\zeta(q-k)$ It looks tough just to start off with. Any ...
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question about property of $L^p$ Lipschitz space

$f\in L^p$ is said to satisfy $L^p$ Lipschitz condition of order $\alpha$ if there exists $C>0$ such that $\displaystyle|h|^{-\alpha}\Big(\int_{\mathbb{R}^d}|f(x-h)-f(x)|^p ...
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An interesting proof using Green's representation formula?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
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88 views

How to construct $\operatorname{End}(V_{\pi})$ using a representation $\pi$

Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning. By the very definition, for ...
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64 views

Fourier transform of power function

Assume that $$\hat f(x)= (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(y) e^{-i\left<x,y\right>} dy$$ is the Fourier transform of a function $f$. What is $\hat f$ if $f(x)=|x|^{2-n}$?
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Could characters in harmonic analysis be generalized into $S^2$?

Consider a locally compact abelian (LCA) group $G$. The start of commutative harmonic analysis is the fact that the collection of characters $\chi : G \to S^1$ (thought of as $S^1 = \mathbb{T} ...
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1answer
112 views

Questions about Fourier Series

I have recently started looking at the topoic of Fourier series. Consider the space of square integrable functions $L_{2}[0,2\pi]$. Where we define the inner product as $(f,g):= \int_{0}^{2\pi}fg dx$ ...
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56 views

Equivalence of Schwartz Space Definition

I've come across two definitions for what it means for a function to be in $\mathfrak S$, the Schwartz space. A function $f \in \mathfrak S$ if $f \in C^\infty$ and for all $j, k \geq 0$ integers, ...
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36 views

Pattern emerging from expansion of sphere in spherical harmonics

Seeing as one can express any function on the sphere in terms of the spherical harmonics, I am interested in what the sphere itself looks like when expanded. To get a function for the sphere, I use ...
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63 views

Definition of Zygmund class

I need help with showing that a function f belongs to the Zygmund class. Only helpful suggestions please, no full solution. I am here to learn for myself. This is work for school (we are allowed to ...
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60 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
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35 views

Poisson integral with discontinuous $U$

Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one for the unit circle). ...
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69 views

About the Fourier transform of the surface measure of the unit sphere

Let $d\sigma$ denote the surface measure on $\mathbb{S}^{n-1}$. To compute its Fourier transform $$ \hat{d\sigma}(\xi)=\int e^{-i x\cdot \xi}\, d\sigma(x), $$ a standard technique (cfr. Folland's ...
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proof that Linear transformation is isometry

let ∑=set of all continous unitary representation and $ Ψ \in{ ∑}$ $π_Ψ: \frac{L^1(G)}{N_Ψ}→ B ( \oplus H_π) $ is definde by $$π_Ψ(f^0)=\oplus π(f) , π \in{ Ψ},f^0\in{\frac{L^1(G)}{N_Ψ} }$$ ...
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62 views

If a function is radial, then its Hardy-Littlewood maximal function is radial as well

I'm looking for a proof of the following statement: $$f\in L(\mathbb R^n)\ \text{ radial } \implies f^* \ \text{ radial}$$ where $f^*$ is the Hardy-Littlewood maximal function defined by: $$f^*(x)= ...
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29 views

Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
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Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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61 views

Why can real variable methods take place of complex variable methods in harmonic analysis' research?

We know that one complex variable methods have been largely used in the early research of (real variable) harmonic analysis. But, as is known to me, there is much difficulty when mathematicians ...
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69 views

Hardy-Littlewood maximal operator

Consider the centered Hardy_littlewood maximal operator $$ \mathcal{M}f(x):= \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| \text{d}y $$ and the uncentered $$ Mf(x):= \sup_{r>0, ...
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Assumptions on the Borel measure in Stein's Harmonic Analysis

I am currently reading the proof of Theorem 1 on Page 13 of Stein's Harmonic Analysis which proves that if $f \in L^{1}(\mathbb{R}^{n})$, then for every $\alpha > 0$, $$\mu(\{x: (Mf)(x) > ...
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58 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
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57 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
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Estimation of trignometric polynomial and Lipschitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
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$\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$

I am trying to show the following: $\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$ I tried to use Fubini's theorem and the fact that ...
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45 views

Convergence of Fourier series in Sobolev space

So the problem is if $f\in H^{\frac{1}{2}}([0,1])\cap C([0,1])$, then $S_Nf$, the partial sum of fourier series converges uniformly to $f$. How would you show this by considering the quantity ...
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Help with biharmonic functions

I tried to use using Green's formula to prove $\triangle u=0$. But it turned out to be wrong because $u=0$ doesn't imply $\triangle u=0$ on the boundary. I am a novice in graduate pde, so I didn't ...
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Local barrier implies barrier?

there. This is part of the textbook of Gibarg's PDE: My question is that how to verify the part in red? How to know $\overline w$ is continous in $\overline \Omega$? Thanks so much! Your help ...
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Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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44 views

Characterization of the Haar measure in terms of the integrals of characters

I was reading a paper and I think that they used the following theorem: Let $G$ compact group and $\mu$ a probability measure on $G$. If $$\hat{\mu}(\xi)= \int_G \overline{\xi(x)} d\mu(x) = ...
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If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 ...
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Find a harmonic function on two concentric balls?

My attempt: I thought about using Poisson Integral formula since the area is two concentric balls. Then I get something like the following: $u(x)=\frac{1}{nw_nR}\int_{\partial ...
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36 views

Harmonic functions that uniformly convergent?

Let $u_k$ be continuous on $\overline\Omega$, $u_k$ harmonic in $\Omega$. Suppose $u_k|\partial\Omega$ converge uniformly. Then $u_k$ converge uniformly in $\Omega$. The hint is using Maximum ...
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Norm of Hardy-Littlewood maximal operator

We define Hardy-Littlewood maximal operator $M$ by \begin{equation} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| dy \end{equation} where $B(x,r)$ denotes the ball centered at $x \in ...
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178 views

Proof of reflection principle for harmonic functions

My attempt: Hi, there! I have known how to prove the above statement when $u\in C^2(U)$, however, I have question about proving the above statement. Because it is $u\in C^2(U^{+}) \cap C(\overline{ ...
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Fourier transform of $f(z)=z^n$ on the unit disk in the complex plane

What is the Fourier transform of the function $f(z)=z^n$, $|z|<1$, and $f(z)=0$ if $z$ is outside of the unit disk.
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integration concerning Fourier transform of homogeneous kernel(of degree 0)

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
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Locally Compact Groups - Reference Request

I start reading an article about locally compact groups $G$ and the group algebra $L^1(G)$,and I need a good book to introduce myself to these concepts. Can you help please? Thanx
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How to use second derivative test?

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 ...
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47 views

When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
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55 views

Convergence of Fourier series in $L^\infty$

So if $f\in L^1(\mathbb{T})$ and $S_Nf\rightarrow f$ in $L^\infty(\mathbb{T})$ ($S_Nf$ is the partial sum of the fourier series of $f$), then $f$ is continuous. How do we show that this is true? In ...
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Estimating the rate of convergence of $|S_Nf-f|$ given that $\|f\|_{H^s}\leq 1$

Given that the Soloblev space norm $$\|f\|_{H^s}^2=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ and the inequality $$\|f(\cdot +\theta)-f\|_{L^2}\leq 2\pi ...
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53 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
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51 views

Wiener Algebra, absolute convergence of fourier series

So how do you prove if $f, g\in L^2(\mathbb{T})$, then $f*g\in \mathbb{A}(\mathbb{T})$. $\mathbb{T}$ denote $[0,1)$ and $\mathbb{A}(\mathbb{T})$ denote the Wiener algebra such that if $f\in ...
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1answer
62 views

Fourier coefficients of a (finite, regular, positive) measure are absolutely summable => the measure has a density

Let $\mu$ be a finite, regular, positive measure on $[0,1)$ such that $\sum_{n\in\mathbb{Z}} |\hat{\mu}(n)| < \infty$. How can I prove that there exists $f(x)$ such that $\mu(dx) = f(x)dx$? ...
3
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1answer
61 views

Question about Hilbert transform(applying plancherel theorem)

Let $f\in S(\mathbb{R})$(Schwartz function on real line). Then Hilbert transform $H$ of $f$ is defined by $\displaystyle Hf(x)=\lim\limits_{t\rightarrow0}\int_{|y|>t}\frac{1}{y}f(x-y)\,dy$ One ...