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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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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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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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 $\mu(dx) = f(x)dx$? ...
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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 ...
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Integral of harmonic function in a ball

Let $f\in C^2(\Omega)$ an harmonic function in $\Omega$, and: $$ \phi(r) = \frac{1}{2\alpha_2r} \int_{\partial B_r(x)} f(y) d \sigma(y) $$ Prove that $\phi '(r)=0$ by calculating the line integral. ...
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23 views

$L^2$ boundedness of Calderon Zygmund operator

A Calderon Zygmund kernel $K$ is a function $K :\mathbb{R}^d-\{0\}\longrightarrow \mathbb{C}$ satisfying, for some constant $B$, 1)$|K(x)|\leq B|x|^{-d}$ for all $x\in\mathbb{R}^d$ ...
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19 views

Determine an operator is weak type (1,1) by its Fourier multiplier

I have an interest on associated Fourier multipler $m$ for a given operator $T$ defined for $f\in L^p(R^d)$, $1\le p<\infty$ by $$ \hat{Tf}(\xi)=m(\xi)\tilde{f}(\xi),\ \ \xi\in R^d $$ where ...
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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 ...
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How to show convergence in $\mathcal{S'}(\mathbb R^{d})$?

We put, $\mathcal{S}(\mathbb R^{d})=$ The Schwartz space and $\mathcal{S'}(\mathbb R^{d})=$ The dual of $\mathcal{S}(\mathbb R^{d})$(The space of tempered distributions). Suppose $\alpha > 1$ and ...
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Reference concerning weak-type (1,1) operator in $\mathbb{R}^d$

I wish someone give me some reference on weak-type (1,1) operator in the d-dimensional Euclidean space. Thanks for any kind help.
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32 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
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Calderón reproducing formula : $\int_{0}^{\infty}\int_{R^d}|\phi_{t}(x-y)||(\phi_t*f)(y)|\frac{dt}{t}dy<\infty$

Suppose that $\int f=0$, $f \in L^2$ and $f$ has a compact support. Let $\phi$ be radial, and such that $\mathrm{supp}(\phi) \in B(0,1)$. Plus, assume that $\int_{R^+} ...
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characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
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Sufficient condition for an operator to be compact in Hilbert space of holomorphic function with respect to Gaussion weight (Fock space).

What I read in a book I could not understand, some one please help. Let $\mathcal{F}=\{f:\mathbb{C^n}\rightarrow\mathbb{C}: \text{$f$ is holomorphic and}\int_{\mathbb{C}^n}\lvert ...
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96 views

Odd form of controlling derivatves

In Muscalu, Schlag - Classical and Multilinear Harmonic Analysis (Cambridge Universitv Press 2013), page 299 there is a rather odd estimate for wich I cannot find any justification: Functions used: ...
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33 views

Is the action on $L^2$ arising from a measure preserving action continuous?

Let $G$ be a locally compact topological group, $X, \mu$ a probability space, and $G\times X \rightarrow X$ a measurable group action which preserves $\mu$ (i.e. $\mu (gA)=\mu(A)$) . Does it follow ...
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22 views

Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
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The variation of Calderon reproducing formula

I'm reading the book 'Classical and multilinear harmonic analysis, Muscalu'. I fail to understand the page 261. Actually, I doubt that the proof is right. Let $f \in BMO(\mathbb{R^d}$) have compact ...
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35 views

Statement regarding mean value theorem for harmonic functions

Suppose $u$ is a function that is harmonic on a domain $D$. Could someone offer a proof of the following statement? $$ u(z_0) = \frac{1}{\pi r^2}\iint_{\{z-z_0\}<r} u(x+iy) \, dx\, dy $$ ...
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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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39 views

Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
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Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family ...
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35 views

TT* Duality argument

I am trying to obtain $L^p$ estimate for a system of nonlinear PDE. I do not have $L^2$ to work with for my problem, I heard $TT^*$ argument is useful tool if one wishes to mapp from $L^p$ to $L^p$. ...
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Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
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293 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
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How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact. Fix $x_{0}\in \mathbb R.$ My Question is : Can we expect to choose, $\phi \in ...
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The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
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41 views

If A has positive Haar measure then $AA^{-1}$ is a neighborhood of $e$

I read the following exercise: Prove that if $G$ is a locally compact topological group with Haar measure $\mu$ and $A \subset G, \mu (A) >0$, then $AA^{-1}$ contains an open neighborhood of the ...
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$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
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The variational formulation of entropy

For $f:\mathbb Z_2^n \to [0, \infty)$, the entropy of $f$ is defined as $$ {\rm Ent}(f) = \mathbb E[f(X) \log f(X)] - \mathbb E f(X) \log(\mathbb E f(X)), $$ where $X$ is a random element of $\mathbb ...
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Why is the derivative of the translates of a measure measurable?

Let G be a topological group and X a measure space. Let $G \times X \rightarrow X$ be a measurable group action, $\mu$ a $\sigma$-finite measure on $X$, and $g\mu$ (for any $g \in G$) the measure ...
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106 views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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Locally compact groups

Let $S= G_1\bigcup G_2 $, where $G_1$ and $G_2$ are two groups. If $S$ is locally compact, is it true that either $G_1$ or $G_2$ is locally compact? Thank you.
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Describing function of a non linearity with memory

Can anyone help me on finding the correct methodology to compute the describing function of the following NL function? ...
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37 views

If $w$ is in weak $A_{\infty}(d\mu)$ where $d\mu$ is a doubling measure, then is $w\,d\mu$ doubling?

Let $\mu$ be a positive Borel measure on $\mathbb{R}^n$ and let it be doubling i.e. there exists a a constant $C>1$ such that $\mu(B(x_0, 2r)) \leq C \mu(B(x_0,r))$ for all balls $B(x_0,r)$. Let ...
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55 views

About p-adic numbers

I'm studying the dual group of the diadic rationals, $\widehat{\mathbb{Z}[1/2]}$, where the dual is the dual of Pontryagin of $\mathbb{Z}[1/2]$. In some papers says that $\widehat{\mathbb{Z}[1/2]}$, ...
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Very simple question regarding sum/difference identity

If I have $\sin(0.7x-47t+C)$ where do I carry my constant $C$? The same with my sum-to-product identities. This problem is showing up for me because I'm studying mechanical waves at the moment. I ...
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Noncommutative Fourier Transform

The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My ...
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Why the $L^1(R)$ space does not have a unconditional basis

Why the $L^1(R)$ space does not have a unconditional basis. It is a known fact that $L^p(R)$ ($1<p<+\infty$) has unconditional basis. A simple example is the dyadic wavelet or Haar system. I ...
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Using normal families to bound a complex integral

I am trying to prove that $$\int_{\partial T(Q)} |F'(z)| \,ds(z) \lesssim \int\int_{T(Q)} |F'(z)| |\varphi'(z)|^2 \log \frac{1}{|z|} \,dx\, dy$$ This is an estimate on page $6$ of this paper by ...
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Building up the Fourier inversion theorem on locally compact abelian groups

I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ...
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Fourier coefficients of a measure and absolute continuity

A relative of a theorem of Peyriere (found in a 1997 paper of Klemes and Reinhold, "Rank One Transformations with Singular Spectral Type") says that if $\mu$ is a Borel probability measure on $S^1$ ...
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An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
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Haar measure on $ \mathbb{R} × \mathbb{T}$ and on dual $\mathbb{R} × \mathbb{T}$

I've solved this exercise somewhat.To complete it please help me Haar measure on G $=$ translation invariant on G $$ μ(A)=μ(A+t)$$ if $ G=\mathbb{R}$ then Haar measure on G is lebesgue measure. and ...
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$\widehat{\mathbb{T}}$ can be identified with $\mathbb{Z}$

$ \mathbb{T} \stackrel{\text{def}}{=} \{ z \in \mathbb{C} : |z| = 1 \}$ $\widehat{\mathbb{T}} \stackrel{\text{def}}{=} \text{Hom}(\mathbb{T},\mathbb{T})$ To show that $\widehat{\mathbb{T}}$ can be ...
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26 views

Local Module Homomorphism

Let $A$ be Banach algebra and $X,Y$ be two left Banach $A$-modules. It is said that the linear bounded map $\phi:X\to Y$ is left $A$-module homomorphism if for any $a\in A$ and any $x\in X$ we have ...
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25 views

Uniqueness in Bochner's theorem

Bochner's theorem : Let $G$ be a locally compact Abelian group. Then for any $ \phi \in \ P(G) $ there is a unique positive Radon measure $ \ μ \in \ $ M ($ \widehat{G} $) such that ...
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70 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ ...
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1answer
34 views

When is a function a Fourier transform of an integrable function?

Specifically, in the case $f(\xi)=\frac{1}{(1+\xi ^2)^\epsilon}$ where $0<\epsilon<1$. I wish to prove this is a Fourier transform of a $L_1$ function. Any insight into the manner would be ...