Questions tagged [harmonic-analysis]

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-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

Harmonic analysis can be considered a generalisation of Fourier analysis that has interactions with fields as diverse as isoperimetric inequalities, manifolds and number theory.

Abstract harmonic analysis is concerned with the study of harmonic analysis on topological groups, whereas Euclidean harmonic analysis deals with the study of the properties of the Fourier transform on $\mathbb{R}^n$.

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Dirac distribution and Sobolev spaces

I see the conclusion that $\delta_{x_{0}} \in H^{s}\left(\mathbb{R}^{n}\right)$ if and only if $s<-n/2$, where $ H^{s}\left(\mathbb{R}^{n}\right) $ is Sobolev space. from many places, and the hint is polar coordinates or Sobolev embedding. How…
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Couterexample to Littlewood-Paley theorem

Let $d\geq 2$ and let $P_j$ be the Fourier multiplier defined on $L^2(\mathbb{R}^d)$ by $\widehat{P_kf}(\xi)=\mathbf{1}_{2^k<|\xi|\leq2^{k+1}} \hat f(\xi)$, for any $k \ge 0$. It has been proven by Fefferman that: Theorem. For any $1
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Show that $T$ is a $(p,p)$-opeator.

Let $\varepsilon>0$, $f\in L^p(\mathbb{R})$ ($1\le p\le \infty$). Opeator $T$ is defined as $$ Tf(x)=\int_{|x-y|>1}\frac{f(y)}{|x-y|^{n+\varepsilon}}\,\mathrm{d}y. $$ Show that $T$ is a weak $(1,1)$-opeator and…
Ryze
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How close can $\mathrm{Ad}(g)x$ be to the identity?

Let $G = \mathrm{SL}(d, \mathbb{R})$ where $d \geqslant 3$. For $g \in G$, define the operator norm of $g$ as $$ \Vert g \Vert_\mathrm{op} := \max \big\{ \Vert gXg^{-1} \Vert_\rho : X \in \mathfrak{sl}(d, \mathbb{R}), \, \Vert X \Vert_\rho = 1…
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Something like fourier transform

Iam a physics student, I don't cover any proper course in harmonic analysis, On the way of studying wavelets I make some guesses, But I don't know they are true or not. They are marked as 1 and 2 1) $f (x)=\int db \psi (x-b) f (b)db $ If this…
ROBIN RAJ
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Can Group of affine transformations generate CWT?

I know group of translations in momentum space generate fourier transform. Because in fourier transform we decompose a function as a sum different frequency components $e^{ikx} $ What group of translations on momentum space do is $U (k) e^{ifx} =…
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Is there a real version of the non-commutative Fourier transform?

In David Applebaum's "Probability on Compact Lie Groups", ch.2 page 36, we have the following definition of the non-commutative Fourier transform where $G$ is e.g. a compact Lie group, $f \in L^1(G; \mathbb{C})$ and $\pi$ is a complex unitary irrep…
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Plancherel theorem on half axis

let $f,g$ be nice. Consider the following integral: $$\int_0^\infty dt\,\int_{-\infty}^\infty dx\, f(x)e^{ixt}\int_{-\infty}^\infty dy\, g(y)e^{iyt}.$$ Fubini's theorem doesn't apply here because $|e^{i(x+y)t}|$ is not integrable on $(0,\infty)$.…
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Fourier transform of $K(x)=\log \left|\frac{1+x}{x}\right|$

I want to calculate Fourier transform of $K(x)=\log \left|\frac{1+x}{x}\right|$, the result is $\widehat{K}(\xi)=c \frac{e^{2 \pi i \xi_{-1}}}{|\xi|}$, where $c$ is some complex constant. Editted: I searched the Math.SE again, it is similar to…
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Derivation of Potential Estimate

I'm reading a book that contains the following statements: Observe that by integrating over the unit sphere the expression: $$u(x) = -\int_0^{\infty}D_ru(x+r\omega)\,dr$$ is valid for all unit vectors $\omega$ in $\mathbb{R}^n$, and by changing…
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Showing a Schwartz Function Bound

I have a question on how to get the correct upper bound of a Schwartz function. Unfortunately, I've never understood this even though I've seen my professors do it a thousand times. I figured it's about time I seek some clarification! Suppose $K >…
Suugaku
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Understanding a particular estimate from Tao's Nonlinear Dispersive equations

On page 339 of Taos's Nonlinear Dispersive Equations, he makes the following estimate (in the context of proving a particular fractional Leibniz rule). Let $M,N$ be dyadic numbers and $P_N$ be the Littlewood-Paley multiplier given by…
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Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the general case these infinite series converge to the…
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Show that the operators given by convolution with the $\sin t / t$ and the distribution p.v. $\cos t / t$ are bounded on $L^{p}(\mathbb{R})$

Question:Show that the operators given by convolution with the smooth function $\sin t / t$ and the distribution p.v. $\cos t / t$ are bounded on $L^{p}(\mathbb{R})$ whenever $1
Lorence
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What is an adapted bump?

In his lecture notes on harmonic analysis, Terrence Tao often uses the notion of an adapted bump. A proper definition is given in Definition 8.14 in these lecture notes. However, I don't really get the essence of this definition, in particular…
Sebastian Bechtel
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