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, Littlewood-...

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300 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. \...
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100 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 ...
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1answer
42 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: ...
3
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1answer
164 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? ...
4
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1answer
57 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 ...
4
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1answer
110 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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1answer
29 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 ...
3
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1answer
64 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 $...
2
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1answer
114 views

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 ...
3
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1answer
83 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 $C^\infty\left(\mathbb{R}\...
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37 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 ...
2
votes
1answer
45 views

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 \...
2
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0answers
21 views

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, u\...
3
votes
1answer
67 views

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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votes
2answers
64 views

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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1answer
27 views

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 $i\in I$...
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1answer
224 views

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 ...
29
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2answers
634 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
32 views

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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0answers
109 views

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 $...
4
votes
1answer
75 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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29 views

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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117 views

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 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\...
2
votes
1answer
32 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 $...
0
votes
1answer
57 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 ...
3
votes
2answers
228 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 ...
3
votes
1answer
63 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 $\...
4
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0answers
55 views

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 = 0,\;\;\;...
2
votes
1answer
41 views

If $\{f_1,f_2,…\}$ a given frame, $T$ a bounded linear, prove that $(Tf,f)=\sum_{n=1}^\infty |(f,f_n)|^2$.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that $$A\|f\|^2\leq\sum_{n=1}^\infty |(...
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1answer
40 views

support of function in topological group

Let $G$ be a compact Hausdorff topological group. Let $U$ be a neighbourhood of the identity $e$, and let $V = U \cap U^{-1}$ where $U^{-1} = \{x^{-1} : x \in U\}$. Apparently there always exists a ...
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56 views

Ideal in Matrix algebra

Let A be a Banach algebra and suppose that $\Lambda$ be a nonempty set. With the following product and norm the matrix space $$M_\Lambda(A)=\{(a_{ij})_{i,j\in\Lambda}:\sum_{i,j\in\Lambda}\|a_{ij}\|<...
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71 views

Is $\Delta^{-1}$ a bounded operator?

Is the inverse Laplacian $\Delta^{-1}: H^{m+2}(M)\mapsto H^m(M)|1$ a bounded operator? Where $M$ is a compact manifold and $H^m(M)|1$ means its elements $f \in H^m(M)$ and $\langle1,f\rangle=\int_{M}...
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1answer
86 views

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 \frac{C}{t}\...
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120 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast f....
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1answer
38 views

Unitary representation on L_1(G)

If $u$ is a unitary representation of a locally compact group $G$ on a Hilbert Space $\mathcal{H}$ then $\pi_u :L^1(G)\rightarrow \mathcal{B}(\mathcal{H})$ given as $$\langle \pi_u(f)\xi\vert\eta\...
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vote
1answer
42 views

Transpose in $ {SL}(2,\mathbb{R})$

Let $SL(2, \Bbb{R})$ denote the group of special invertible $2\times 2$ matrices over $\mathbb{R}$. As a locally compact group, the Haar measure of $SL(2, \Bbb{R})$ is computable through Iwasawa ...
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Existence of Inverse Laplace Transform

Let $F(s)$ the Laplace transform of a function $f(t)$ . Under which conditions on $f(t)$ there exist a unique $g(t)$ such that $g(t) = \mathcal{L}^{-1}\{e^{- F(s)}\}(t) \quad $ ? ($\mathcal{L}^{-1}$...
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1answer
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Can the 0 element of the Fourier Algebra be represented as a coefficient function of two non-zero vectors?

Consider an infinite discrete group $G$, and its associated Hilbert space $l^{2}(G)$. For $t\in G$, let $\lambda(t):l^{2}(G)\to l^{2}(G)$ denote the map $[\lambda(t)x](s) = x(t^{-1}s)$. That is, $\...
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42 views

A proposition to prove the real interpolation of positive exponent

Let $p,A\in(0,\infty)$ $\|f\|_{L^{p,\infty}(X,\mu)} := \sup \{\lambda\mu(\{|f|\geq\lambda\})^{\frac{1}{p}}\}$ Show that the following are equivalent (1) $\|f\|_{L^{p,\infty}(X,\mu)}\leq C_pA$ for ...
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31 views

Prove $\hat{f}(\omega)\neq 0$ if $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete

Let $f\in L^1(\mathbb{R})$. s.t $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete. Prove that $\hat{f}(\omega)\neq 0$ for all $\omega\in\mathbb{R}$ Suppose the system is complete for any $g\in L^1(\mathbb{R}...
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43 views

Proving $\mu\ast K_n\to\mu$

Let $\{K_n\}$ be approximating unit and $\mu\in M(\mathbb{T})$. Show that $\mu\ast K_n\to \mu$ weakly means $$\int f(t)d(\mu\ast K_n)(t)\to\int f(t)d\mu(t)$$ Suppose $\mu\in L^1(\mathbb{T})$, I know ...
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vote
2answers
31 views

Proving norms inequality for fourier transforms

Let $F\in PW_\pi$. Prove that $\Vert F^\prime\Vert_{L^2(\mathbb R)}\le\pi\Vert F\Vert_{L^2(\mathbb R)}$ I know that the derivative of $F=\hat{f}$ is given by the formula $F^\prime(\omega)=\widehat{-...
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votes
1answer
44 views

Calculate $\mathscr{F}((1+t)^{-3})$

Let $$f(t)=\cases{\frac{1}{(1+t)^3}&t>0\\0&t<0}$$ Does: a.$\hat{f}$ is differentiable? b.$\hat{f}\in L^1(\mathbb{R})$? c.$\hat{f}\in L^2(\mathbb{R})$? Seems like we need to calculate ...
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33 views

Proving a sum is finite using Equidistribution

Let $\phi:\mathbb{R\to R}$, be an integrable function with finite integral on $[0,1]$($\int_{[0,1]}\phi(x)dm<\infty$) and $\phi(x)=\phi(x+1)\forall x\in \mathbb{R}$. Prove that $$f(x)=\sum_{n=1}^\...
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votes
1answer
60 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 ...
1
vote
1answer
47 views

Differential operator that must be a constant multiple of the Laplacian

Can you help me show that if $L$ is a partial differential operator given by $$L=a{{\partial^2}\over{\partial{x^2}}}+b{{\partial^2}\over{\partial{y^2}}}+c{{\partial^2}\over{\partial{x}\partial{y}}}$$ ...
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132 views

What are single layer and double layer potentials?

I have never encountered the terms "single layer potential" and "double layer potential" in my (under)graduate studies, although there was a firm development of partial differential equations and some ...
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61 views

Gradient curve of a harmonic function

I am reading the paper "Energy of Harmonic function and Gromov proof of Stalling theorem" https://www.math.ucdavis.edu/~kapovich/EPR/energy.pdf I have no clue about the lemma 8.4(i). What is gradient ...
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0answers
38 views

Group $C^*$-algebra elements as limit of self-adjoint integrable functions

Assume $G$ is a locally compact abelian group and let $C^*(G)$ denote its group $C^*$-algebra. I am reading a proof that uses the 'fact' that some $f\in C^*(G)$ is a limit of self-adjoint functions $...
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1answer
24 views

How can I prove that $\max_{\partial K}\varphi$ does exist?

Let $\varphi:\Omega\to[-\infty,+\infty[$, where $\Omega\in\Bbb C$ is a domain and $\varphi$ is upper semicontinous, i.e. $\varphi(z_0)\ge\limsup_{z\to z_0}\varphi(z)\;\;\;\forall z_0\in\Omega$. How ...