# Tagged Questions

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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### Riemann Lebesgue Lemma application?

Riemann Lebesgue Lemma. shows that if $f \in L^1 ( \bf R)$ then the Fourier transform of $f$ goes to $0$. Does this also implies that $f(x) \to 0$ as $\vert x \vert \to \infty$
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### Homogeneous distribution

In Wikipedia, it says The Dirac delta function is homogeneous of degree −1, with the following formula: However, I can not understand why the last equality is true. Can someone show me the detailed ...
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### Can the real part of an entire function be bounded above by a polynomial?

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function such that $Re(f)\le |p(z)|$ for some polynomial, can we derive that $f(z)$ is a polynomial. If $p(z)$ is constant, then this can be shown by ...
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### Do we have for $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ exists for $u$ subharmonic?

Let $u:\mathbb{C}\to \mathbb{R}$ be a subharmonic function. Do we have that the limit $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ converges to a (possibly infinite) ...
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### Number of copies of irreducible unitary representation in $L^2(G)$ for compact group $G$?

Peter-Weyl Theorem is concerned with expressing $L^2(G)$ as closure of direct sum of subspace generated by irreducible unitary representation. Every irreducible representation on a compact group is ...
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### references for abstract harmonic analysis

I am currently studying a basic course in abstract harmonic analysis. I do have a good background in abstract algebra and functional analysis but I have not done a course in Fourier analysis. Is it ...
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### 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 ...
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### Strichartz Estimate with Fourier Transform

Let $f$ be a Schwartz function. Prove that, whenever $2\le r < \infty,$ $$\| e^{it \Delta} f\|_{L^{3r}(\mathbb{R}^2_{xt})} \le c \| \widehat{f}\|_{r'},$$ Where $1/r + 1/r' = 1.$ My Attempt My ...
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### Is the composition of an harmonic function with an analytic function an harmonic function in any dimension?

I was wondering if it is true or not that, given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ and $g:\Omega\subset\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that $f$ is harmonic and $g$ real ...
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### Examples of non unimodular groups

I'm looking for criteria to know if a (lie) group is unimodular, the thing is that I only know one non unimodualr group that arrises naturaly, the affine group ax+b, and their direct generalizations. ...
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### Commutator Characterization of $BMO(\mathbb{R})$

Let $a:\mathbb{R}\rightarrow\mathbb{C}$ be a locally integrable function, and let $H$ denote the Hilbert transform. Suppose that the commutator operator $[a,H]$ defined by $[a,H]f:=aH(f)-H(af)$ is ...
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### An operator is linear and bounded on a hilbert space

an operator linear and bounded on a hilbert space Let H be the Hilbert space L^2(R), and assume that the continuous function g satisfies 0
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### Foundation Semigroups Examples

I recently started looking into the subject of foundation semi-groups and I'm trying to find simple (none-trivial) examples of foundations semi-groups. I know that every locally compact group is ...
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### Extension to a continuous linear functional

Extension to a continuous linear functional May this functional be extended to a continuous linear functional on these Hilbert spaces?
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### Proving Div-Curl Lemma through Paraproducts

I want to to prove the classical div-curl lemma of Coifman-Lions-Meyer-Semmes in the following form: Div-Curl Lemma. Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a Schwartz function such ...
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The support of a function is the closure of the set of points where the function has non zero values. The function $f(x)=\cos(x)$ is zero only at the points $x=\frac{(2k+1)\pi}{2}$, $k \in ... 1answer 24 views ### Properties of the Fejer kernel The Fejer kernel$k_m : \mathbb{R} \to \mathbb{C}$is defined by$k_m (t) = \frac{1}{2\pi (m+1)} \sum^m_{n=0} \sum^n _{k=-n} e^{ikt}$One of the properties of the Fejer kernel is For any$\delta ...
In Notes 4 on regularity of harmonic maps, T. Tao asserts the following lemma, which in short, allows us to localize a map $u:\Omega\rightarrow S^{m-1}$ (here, $\Omega\subset\mathbb{R}^{2}$ is a ...
Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all \$j \in ...