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Results tagged with harmonic-analysis
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user 272040
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.
3
votes
The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
Heuristics
Since $e^{-|x|^a}$ behaves like $1-|x|^a$ plus better behaved terms close to zero, and since formally the Fourier trasform of $|x|^a$ is proportional to $|\xi|^{-a-n}$, I would expect that …
- 1,065
5
votes
1
answer
502
views
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ i...
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its elements have …
- 1,065
6
votes
Accepted
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ i...
Edit. For the sake of improving the quality of the post, I modified the proof to make it work for all $M>n$ after Prof. Tao’s comments (the previous version was admittedly way too loose).
In the end I …
- 1,065