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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What could be the reasons due to which we avoid Fourier Transform and must choose other transforms?

I am fond of Fourier series & Fourier transform. Fourier series formula for any periodic signal g(t) is given as Fourier Transform of aperiodic signal x(t) is given as In Fourier domain, ...
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21 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, ...
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20 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 $[0,1]\times[0,1]$.
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What the terms “basis functions” and “orthogonal” denote in the case of signals?

I am a beginner. I have read that any given signal whether it is a simple or complex one, can be represented as summation of orthogonal basis functions. Also, there are many ways through which basis ...
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44 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 ...
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$L^{p}$ 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 ...
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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 ...
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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 ...
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28 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 ...
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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 $c=0?$
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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 ...
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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 ...
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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. ...
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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 ...
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43 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 ...
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35 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: ...
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77 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 ...
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15 views

What does it mean for a subset of $L^{\infty}(G)$ to separate the points of $G$?

Let $G$ be a locally compact group, and $M\subset L^{\infty}(G)$. I would naively define $M$ to separate the points of $G$ if for all $s,t\in G$, there exists $f\in M$ such that $f(s)\neq f(t)$. But ...
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1answer
31 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 ...
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325 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 ...
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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 ...
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1answer
37 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? ...
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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 ...
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48 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 ...
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51 views

Solutions of the constant coefficient Helmholtz equation via the Fourier transform

When $f$ is a rapidly decaying Schwartz function, $$ g(x) = \frac{1}{2\lambda} \int_{\mathbb{R}} \sin \left(2\lambda\left|x-y\right|\right) f(y)\ dy $$ is an element of ...
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Lebesgue differentiation theorem for Orlicz spaces

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow ...
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2answers
536 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 ...
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1answer
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For Dirichlet's problem on the unit disk, show that the solution is harmonic.

I am trying to prove that $u(re^{i\theta}) = \sum\limits_{n=-\infty}^{\infty}\hat{f}(n)r^{|n|}e^{in\theta}$ is the solution to the Dirichlet problem on the unit disk if on the boundary of the unit ...
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32 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=C$\ {0, 1}. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0log|z|−a_1log|z−1|$$ is the real ...
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33 views

A linear functional of $H^k$

Define $H^k$ as usual: $$H^k=\left\{f; \int (1+|\xi|^2)^k|\widehat{f}(\xi)|^2d\xi<\infty\right\}$$ Define $l(u)=\int (1+|\xi|^2)^k\widehat{u}(\xi)d\xi$, how to show that $l\in (H^k)^*$. I tried ...
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1answer
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Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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Norm convergence of approximations to the identity

Let $\varphi \in L^1(\mathbb{R}^d)$ be such that $\int_{\mathbb{R}^d} \varphi(x) \, dx = 1$. For each $\varepsilon>0$, let $\varphi_\varepsilon:= \varepsilon^{-d} \varphi\left( \dfrac x ...
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Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
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inserting absolute value in Hilbert transform and a discrete version of Hilbert transform

It is well known that the Hilbert transform $H(f)(x)=p.v. \int\frac{f(x-y)}{y}dy$ is bounded on $L^p(\mathbb{R})$ for $p\in(1,\infty)$. I want to consider some variants of $H$. 1) What happens if we ...
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Application of Plancherel/Parseval

Assuming $u,v\in L^1\cap L^2$, then how do you show that $$\int uv=\int \hat{u}\hat{v}$$ I tried using Plancherel, but didnt give any nice result. Any ideas/hints? Thanks
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Iterated Limits Along an Ultrafilter

Setting: Let $\mathfrak{U}$ be an ultrafilter on an index set $I$. Let $G$ be a compact group with identity $e$, and let $\mathbb{T}$ denote the unit circle in the complex plane. For each $i\in ...
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Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
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1answer
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Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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Decay of the Fourier transform of the surface measure of the sphere via uncertainty

I'm working through Tao's Recent Progress on the Restriction Conjecture notes (http://arxiv.org/abs/math/0311181). Currently, I'm working on problem 2.4, which will eventually allow us to compute the ...
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Inverse Fourier transform of cut off of Fourier transform

Suppose we have a function $f(x)$ such that $$|\frac{d^n}{dx^n}f(x)| \leq C(1 + |x|)^{-n}$$ Take the Fourier transform $\hat{f}(\xi)$ and consider the function $g(\xi) = \chi(\xi)\hat{f}(\xi)$, where ...
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Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
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A question about a proof in Lang's $SL_2(\mathbb{R})$

The following is a lemma in Lang's book $SL_2(\mathbb{R})$. It's the last line of the proof that I don't understand. Let $G=SL_2(\mathbb{R})$ , $E$ a Banach space, and let $\pi$ be an irreducible ...
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1answer
31 views

Why is boundedness of the ball multiplier equivalent to the convergence of Fourier transform in Lp?

Let $\mathcal{F}$ be the fourier transform operator and let $T_R$ = $\mathcal{F}^* \chi_R\mathcal{F}$ where $\chi_R$ is the indicator function on the ball of radius $R$. Hence $T_R$ is the fourier ...
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Function decomposition in harmonic analysis

In the book "Weighted Norm Inequalities and Related Topics" by José García-Cuerva, J.-L. Rubio de Francia page 144 it was shown that for a measurable function $f$ and $t>0$ $$ |E_t|\leq ...
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45 views

A question from Stein's Harmonic Analysis - real variable methods' book.

In the book of Stein, on page 9: "Let us remark that these additional properties easily lead to the following conclusions among others. First note that $\mu (B) >0$ for any ball $B$, which is a ...
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1answer
33 views

Uniform Approximation by Finite Wavelet Sum

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the ...
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1answer
28 views

Closedness of set in product topology

Let $A$ be an abelian group and let $\mathbb{T} \subset \mathbb{C}$ be the unit circle. For $a,b \in A$, let $M(a,b) \subset \mathbb{T}^A$ be the set of all functions $f: A \to \mathbb{T}$ such that ...
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1answer
45 views

Maximal Function Estimate

Suppose $\psi$ is a rapidly decreasing function; i.e. for all $N>0$ there exists a constant $C_{N}$ such that $\left|\psi(x)\right|\leq C_{N}(1+\left|x\right|)^{-N}$. Define a family of functions ...
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What classical conditions give unique Laplace equation solutions on a half-plane?

Solutions of the Laplace equation on the upper half plane of $\mathbb{R}^{2}$ are not unique: $$ \frac{\partial^{2}}{\partial x^{2}}u+\frac{\partial^{2}}{\partial y^{2}}u = ...