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Results tagged with harmonic-analysis
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user 37103
Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
5
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Chapter X, Section 2, Proposition 1 of Stein's harmonic analysis
The Proposition claims:
Suppose we are given a countable collection $\{d\mu_j\}$ of finite nennegative measure on $\mathbb R^n$, supported in a fixed compact set. Define the maximal operator
$$ Mf(x …
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-1
votes
1
answer
152
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$\ell^q$ analog of square function
It is a classical result in harmonic analysis that
$$ \|\|P_kf\|_{\ell^2_k}\|_{L^p_x}\approx\|f\|_{L^p} $$
for $p\in(1,\infty)$, where $P_k$ is the Littlewood-Paley decomposition onto frquency $\app …
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8
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1
answer
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Cricket and the Hardy-Littlewod maximal function
I'v read somewhere that one motivation for Hardy to define his maximal function is the game of cricket. But I can't see how they are related. Could anyone provide some more information on their connec …
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2
votes
1
answer
809
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Decoupling in mixed norm spaces
Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|_{L^p}$ into an $L^2$ sum of $\|f_\theta\|_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where $\the …
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2
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0
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394
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Hölder-Zygmund spaces of negative order
In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the sa …
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3
votes
Accepted
Does the induced representation preserve norm?
No. Let $G=S_3$ generated by 3-cycle $x$ and involution $y$. Let $N=A_3$. Let $ x$ act on $X=\mathbb C$ by multiplication by the third root of unity $\omega$. Let $a=1+ix$. Then $\pi(a)$ has norm $|1+ …
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9
votes
1
answer
903
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The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?
I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a …
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2
votes
Show that the Laplacian operator on the Heisenberg group is negative
The factorization of the Heisenberg Laplacian is well known, see
http://www.scirp.org/journal/PaperDownload.aspx?paperID=22807
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Infinitely many independent functions that are only frequency localized?
For all $a\in(0,K)$, the function $\hat 1_{[0,a]}$ is $K$-frequency localized, yet the function itself decays like $1/x$ as $x\to\infty$ (as easily seen by integration by parts), so it is not spatiall …
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Accepted
$L^2$ function in Schwartz space?
This is quite ad hoc, but your integral equation can be reverse engineered to a differential equation (Maybe this is what you really want to solve?)
$$ \partial_t\varphi=\alpha(t)f\varphi,\ \varphi(0 …
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