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Results tagged with fourier-analysis
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user 1044
The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.
3
votes
Fourier transform of $\exp(-\|x\|_p)$: more general question
I don't have an actual answer for you, but I'll give you some possibly helpful buzzwords to try to google your way to an answer: stable random vectors. As I understand it (at least in the case $p=2$) …
- 11.4k
11
votes
Accepted
Fourier transform of $\exp(-\|x\|_p)$: more general question
Okay, I think I do have an answer now. I'm borrowing arguments from the proof of Lemma 2.27 in the book "Fourier Analysis in Convex Geometry" by A. Koldobsky (apparently not available online at all). …
- 11.4k
5
votes
Accepted
In what ways is the standard Fourier basis optimal?
Here's something similar to what you conjecture. Let's work instead with functions $\mathbb{R} / 2\pi \mathbb{Z} \to \mathbb{C}$; one can do similar things in the real-valued case but I find the comp …
- 11.4k
1
vote
Berry Esseen inequality for multidimensional distributions
There are many results along those lines in Bhattacharya and Rao, Normal Approximation and Asymptotic Expansions.
- 11.4k
1
vote
Accepted
Convergence of Fourier series for $C^p$ functions
There is a theorem of Lebesgue that says that for a continuous periodic $f$,
$$
\|f - S_N f\|_\infty \le C \log N \|f - f^* \|_\infty.
$$
This appears as Theorem 2.2 in Rivlin's book. Combined with t …
- 11.4k
3
votes
Approximate a probability distribution by moment matching
In the context of 3), what I have heard from folklore is that when (1) holds, the Kolmogorov distance (not total variation) is bounded by something like $1/\sqrt{m}$. This bound follows if (1) holds …
- 11.4k