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, ...

learn more… | top users | synonyms

1
vote
1answer
23 views

Example of function not Fourier Invertible in $L^1$

It is well known that, if $1 < p \le 2$, then, for every $f \in L^p$, $$ \int_{[-R,R]^n} e^{-2\pi i x \cdot y} \hat{f}(y) dy \rightarrow f(x) $$ As $R \rightarrow \infty$, in the $L^p$ sense. ...
5
votes
1answer
48 views

If $\gamma$ is irrational, then $\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)\to \int_T f(t)\,dt$

I need to show that $$ \lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)=\int_T f(t)\,dt. $$ Here $\gamma$ is any irrational number on the real line and $f(t)$ is any continuous periodic ...
0
votes
0answers
21 views

Properties of Certain Example of Nonuniqueness to Heat equation

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
2
votes
1answer
21 views

Existence of Certain Locally Integrable Function Defining a Tempered Distribution

We say that a distribution $T$ is tempered if for every sequence $\left\{\phi_{n}\right\}$ in $C_{c}^{\infty}(\mathbb{R}^{n})$ tending to $0$ in the topology of the Schwartz space ...
1
vote
1answer
26 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...
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 ...
3
votes
1answer
52 views

$L^{p}$ Boundedness of Fourier Multiplier without Littlewood-Paley

Suppose $\zeta\in\mathcal{S}(\mathbb{R}^{n})$ is a Schwartz function such that $\widehat{\zeta}$ is compactly supported away from the origin (say in the annulus ...
6
votes
1answer
276 views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
4
votes
1answer
106 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
4
votes
1answer
87 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
2
votes
0answers
38 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
3
votes
1answer
30 views

Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
1
vote
0answers
18 views

Continuity of Translation and Dilation on $L^p$ spaces

Let us consider any $f \in L^p(U)$, where $U \subset \mathbb R^n$ is open, and $1 < p < \infty$. We know the translation operator $f(x) \mapsto f(x+a)$ and the dilation operator $f(x) \mapsto ...
11
votes
1answer
338 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 ...
2
votes
0answers
16 views

Existence of Fourier Transform for Implicit function

Given an "explicit" function $f:\mathbb{R}^n\to\mathbb{R}^n$, (e.g $F(x_1,\dots x_n)=\cos(x_n)+x_1^2e^{x_2}$) under some assumptions one can allegedly develop a Fourier transform given by ...
3
votes
1answer
59 views

How to compare the Hardy-Littlewood maximal function for balls and cubes?

I am currently working through a set of notes I found on the internet at: http://math.msu.edu/~charlesb/Notes/DuoChapter2.pdf I am up to page 8, and the Hardy-Littlewood maximal function for balls ...
0
votes
1answer
14 views

Showing a function is harmonic on a domain - Imaginary part of $(A\cosh(z)+\frac\pi z)$

How to know that $\text{Im}(A\cosh(z)+\frac\pi z)$ is harmonic on domain $\{z|0\lt\text{Im }z\lt \pi\}$ where $A\in\Bbb R$? I am not sure how I would verify Laplace's equation here(which I imagine is ...
0
votes
1answer
23 views

Neumann problem on $\Omega$. Does $U$ solving the problem imply $U+c$ does?

Let $U$ solve the Neumann${}$ problem${}$ for laplace's equation on a${}$ domain $\Omega$. Show that $U+c$ also solves this problem for any $c\in\Bbb R$. What is being asked of me? Does this mean ...
1
vote
1answer
35 views

integrating something from a partial derivative $v=\int \frac{2x}{x^2+y^2}\,dy$

i am trying to learn harmonic analysis, and i have$$\frac{\partial u}{\partial x}=\frac{2x}{x^2+y^2}=\frac{\partial v}{\partial y}$$ and i want to get $v$. so what i do is: $$v=\int ...
0
votes
0answers
27 views

Harmonic Analysis of Finite Groups

If I understand correctly, the basic goal of harmonic analysis on finite groups is to find isotypical subspaces of a given set. Why is it important to do so? What are the advantages of decomposing a ...
1
vote
2answers
17 views

An upper semicontinous function which is not subharmonic.

Let $\Delta\subset\Bbb C$ be the open unitary disk. Let $\varphi:\Delta\to\Bbb R$ defined as follows: $\varphi(z)=1$ if $\Re z\ge0$, $\varphi(z)=0$ otherwise. So $\varphi$ is upper semicontinous. In ...
1
vote
1answer
71 views

Have some queries about Fourier Transform

I have some queries about the Fourier transform In most of the cases, the Fourier transform of a signal is symmetric with respect to positive and negative frequency. I think the computational ...
0
votes
0answers
31 views

do (everywhere) continuity and linear directional derivatives imply differentiable?

Let $f: \mathbb{R}^m\rightarrow \mathbb{R}$ be an everywhere continuous function and suppose there exists a linear map $L: \mathbb{R}^m\rightarrow \mathbb{R}$ such that $$\lim_{t\rightarrow 0} ...
1
vote
1answer
23 views

Jacobian for Partial Iwasawa Coordinates

I am working through Terras' Harmonic Analysis, V2, and am stuck on I believe a notational point. We are asked to show that for ...
0
votes
1answer
26 views

Convergence of $S_{n}(f;t)$

Let $f \in L^P(T)$ for some $p>1$. If $n$th Fourier partial sum $S_n(f;t)$ converges almost everywhere as $n \rightarrow \infty$, does the limit have to be $f(t)$ almost everywhere? I am trying to ...
4
votes
3answers
73 views

What the terms “basis functions” and “orthogonal” denote in the case of signals?

I am a beginer. I have read that any given signal whether it is simple or complex one,can be represented as summation of orthogonal basis functions. Here, what the terms Orthogonal and Basis function ...
0
votes
0answers
4 views

Long time statistics of random functions

I'd like to understand if an average over random functions can be approximated with a Markov process in the long-time limit. Let $$ X_t = \sum_k a_k \cos(\omega_k t + \phi_k) $$ a random function, ...
2
votes
2answers
94 views

What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

Let us take an example, a white ray (which is composed of bunch of frequency components) is passed through a prism, the ray gets split (decomposed) into its elementary vibgyor colours (i.e.different ...
2
votes
0answers
24 views

the convolution of integrable functions is continuous?

I just look through a webpage from "mathoverfolw", on http://mathoverflow.net/questions/136681/the-convolution-of-integrable-functions-is-continuous Let me remark that it is sufficient that one of ...
0
votes
0answers
10 views

Is this function continuous on topological groups?

$G$ is a locally compact Hausdorff topological group, $m$ is a (left) Haar measure on $X$, $ 0<m(A)<+\infty $, $0<m(B)\leq +\infty $. Let $f:G\longrightarrow R $, $f(x) = \int_G ...
2
votes
0answers
36 views

Does a Plancherel Style Theorem for the Hardy Space $\mathbb{H}^2$ on the Unit Circle Exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathbb{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
0
votes
1answer
22 views

Existence of invariant mean for $\mathbb{R}^n$

I am working on exercise 2.3(b) from "Essential Results of Functional Analysis" by Zimmer and couldn't find a similar question asked here. A mean $m$ on a measure space $(X,\mu)$ is a continuous ...
1
vote
1answer
31 views

Inequality for Ratio of Hardy-Littlewood Maximal Function over Balls and Cubes

Let $M$ denote the centered Hardy-Littlewood maximal function using balls, and let $M_{c}$ denote the centered Hardy-Littlewood maximal function using cubes. Exercise 2.13 in [L. Grafakos, ...
0
votes
0answers
23 views

Prove one property of harmonic function

Let $u(x)$ be a harmonic function defined in the square $[0,1]\times[0,1]$. Suppose that $u(x_k)=0$, where $x_k=(1/k,1/k).$ Prove that $u(x)=0$ everywhere in $[0,1]\times[0,1]$.
0
votes
1answer
52 views

Haar measure on locally compact group

Please I need a help to solve two problems in the book of principles of Harmonic analysis of Deitmar and Echterhoff Exercise 1.4 Let $G$ be a locally compact group with Haar measure $\mu$, and let ...
0
votes
0answers
21 views

Eigenfunctions of the Fourier transform on locally compact abelian groups

The eigenfunction theory of the Fourier transform on $\Bbb R$ is well-understood. For example, the Hermite-Gauss functions are eigenfunctions with eigenvalues $i^n$; in fact, this comprises the ...
1
vote
1answer
29 views

Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
5
votes
1answer
22 views

A property of a ideal of Banach algebras

Let $B$ be a Banach algebra and $A$ be a bi-ideal of it. Suppose that for any $b\in B$, $Ab=\{0\}$ implies $b=0$. Now could we say that for some $c\in B$ if $cA=\{0\}$ then $c=0?$
4
votes
0answers
93 views

When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
3
votes
0answers
25 views

Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Explain the geometrical interpretation a pair of harmonic function conjugated each other. Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with ...
1
vote
0answers
23 views

Intuition Behind the Riesz Transform

Define the Riesz transform in singular integral form \begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. ...
0
votes
0answers
28 views

Please help me with Fourier series problem!

(a) Find Fourier series of $f(x)$ on $[-L,L]$ $$f(x)=\begin{cases} x(L-x) & 0\le x<L \\ x(L+x) & -L < x < 0 \end{cases} $$ (b) Find $f'(x)$ and $\int_{-L}^x f(x)\, dx$ and the ...
0
votes
0answers
45 views

fourier Series gone Bad

I understand why the series converges uniformly for $\lambda=\frac{2}{\pi}$ and i can get a very non elegant answer for a( as An=0 and Bn is a verly clumsy expression): the series as is: I ...
0
votes
0answers
30 views
0
votes
1answer
36 views

Solve Intergrals Using Inverse Fourier Transform

a)Find f(x), the insverse fourier transform of F(ω) b) Does the fourier transform of f(x) equal to F(ω)? c)use your answers to calculate these Integrals: if I'm not mistaken the answer to a is: ...
4
votes
1answer
80 views

An inequality for a maximal function on an $n$-ball.

We have $Mf(x) = \sup_{r>0} \frac{c_n}{r^n} \int_{|y|\le r} |f(x-y)| dy$ the maximal function, where $r^n/c_n$ is the volume of the n-dimensional ball of radius $r$, $|y|\le r$. I want to show ...
0
votes
1answer
35 views

Harmonic analogue of the Weierstrass approximation theorem

The Weierstrass approximation theorem says that, given any continuous function $f(x)$ on a closed interval, there is a polynomial which approximates it arbitrarily closely. I'm looking for a theorem ...
4
votes
1answer
38 views

A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
1
vote
1answer
43 views

Pontryagin Dual of the Unit Circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
3
votes
1answer
50 views

Showing the range of Fourier transform on $L^1(\mathbb{R})$ is in $L^1(\mathbb{R})$ for a particular scenario

This is a problem from Katznelson's Harmonic Analysis book(Page 143, Problem 8): Suppose $f\in L^1(\mathbb{R})$ and is continuous at $0$. Also suppose $\hat{f}\geq 0$, then show that $\hat{f}$ is in ...