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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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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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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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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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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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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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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77 views

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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29 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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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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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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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 ...
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1answer
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Inequality between Fourier coefficients implies inequality for $L^p$ norms on the circle

Given two functions from $L_p [-\pi,\pi]$, where $p\geq 2$, $p$ is an even integer, and $f_n>|g_n|$ for every $n$ (where $f_n$ is the $n$th Fourier coefficient), I need to prove that ...
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Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
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Locally compact group and continuous function with compact support

Suppose that $G$ is a locally comact group, let $H$ be a open subgroup of $G$ and let $\phi:G\to\Bbb C$ be a continuous function such that $Supp\phi=:\overline{\{x\in G: \phi(x)\neq 0\}}$ is compact. ...
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Open subgroup and group algebra

Let $G$ be a locally compact group and $H$ be an open subgroup of $G$. Consider the group algebras $L^1(G)$ and $L^1(H)$ with convolution product and consider $L^1(H)$ as a subalgebra of $L^1(G)$ ...
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Discontinuous characters on locally compact abelian groups

I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither ...
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Weak* convergence in $(L^q)^*$ and convergence in $L^p$

Let $\{\phi_n\}$ be a sequence in $L^p(X)$. Assume that $\phi_n\to \phi$ weak* under the natural identification $L^p\cong (L^q)^*$. Of course, it is not true in general that $\phi_n\to \phi$ in $L^p$ ...
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I am having a hard time understanding just how quotients such as $\mathbb{T}$ are constructed.

I encountered the quotient $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ while beginning to study harmonic analysis. Despite having taken elementary abstract algebra and number theory, I am still having ...
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Fourier transform of a potential

I need help computing the distributional inverse Fourier transform of the function $1/|x|^2$ in dimension two. The integral makes sense written as \begin{align} 1/2\pi \int_{\mathbb{R}^2} e^{ix\xi} ...
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Using Plancherel's Theorem to Prove the Gauss Sum

I'm interested in proving the following: Where $p$ is an odd prime and $z$ is a primitive $p$th root of unity, we let $Q(p)=\sum^{p−1}_{k=0}z^{k^2}$. Prove: $|Q(p)|=\sqrt{p}$. Specifically, I want ...
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non degenerate representation

Let π:G →L(H) be a unitary representation Then the map π1: $\mathbb L^1(G)$→L(H) is nondegenerate. where ’nondegenerate’ is meant in the sense that For every non-zero $\xi \in \mathcal{H}$ ...
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A questions about the schur's lemma

Schur's lemma is this: If (ρ1,V1) and (ρ2,V2) are irreducible representations of a group G, then any nonzero homomorphism ϕ:V1↦V2 is an isomorphism. or Schur's Lemma. a. A unitary ...
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If the Fourier transform of a probability measure goes to zero at infinity, can the measure have a point mass?

Let $\mu$ be a probability measure on $\mathbb{R}$. Is the following implication true? $$ \widehat{\mu}(y) \rightarrow 0 \text{ as } |y| \rightarrow \infty \quad \Rightarrow \quad \mu(\{x\})=0 \quad ...
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Is every unitary representation a direct sum of irreducible subprepresentations?

I've read that any unitary representation of a compact group decomposes as a Hilbert space direct sum of irreducible representations. In the book I'm reading this is stated as a prong of the ...
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Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
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Kernel of a representation and central projections

Let $G$ be a locally compact group, and $B(G)$ be the Fourier-Stieltjes algebra of $G$. I'm trying to see why the following is true: let $\pi$ be a representation of $G$, the the kernel of $\pi$ in ...
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Why are all Laplacian eigenfunctions on the square obtained by separation of variables?

Let our domain $\Omega = (0,2\pi)\times(0,2\pi)$ be a square with sides parallel to the axes. Consider the Dirichlet eigenvalue problem $\Delta u+\lambda u=0$ with Dirichlet boundary condition $u=0$ ...
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Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
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Application of weak $L^p$ estimate besides for proving boundedness of some linear operator

For all $1\leq p< \infty$, weak-$L^p(\mathbb{R}^d)$ space is defined as a set of all functions $f$ such that $$\gamma^p|\{x\in \mathbb{R}^d: |f(x)|>\gamma\}|<\infty$$ for every ...
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functions of positive type

We say that $∅∈\mathbb L^∞(G)$ is of positive type, iff $$ \iint f(x)\overline{f(y)} \phi(y^{-1} x)\;dy\;dx≥0 $$ for all $f∈ \mathbb L^1(G)$ The set of all continuous functions of positive type ...
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Computing the modularity function

Let B be the group of real matrices of the form \begin{pmatrix} 1 & x\\ 0 & y\end{pmatrix} with y ≠ 0. Then the modular function Δ is given by $$Δ\begin{pmatrix}1 & x \\ 0 & ...
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*-homomorphism and *-representation

I understand the concept of the unitary representation of G . A unitary representation of G is group homomorphism π:G→U(H) where H is a complex Hilbert space and U(H) is the group of unitary operators ...