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Results tagged with fourier-analysis
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user 172051
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.
2
votes
0
answers
149
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An oscillatory integral
Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates?
\begin{equati …
- 351
3
votes
2
answers
397
views
A Sobolev embedding theorem for functions on spheres
$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}
\ …
- 351
0
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A Sobolev embedding theorem for functions on spheres
This inequality may be proved by a calm calculation. We can assume that $f$ supports near $(0,\cdots,0,1)\in \mathbb{S}^{d-1}$. Denote the first $d-1$ coordinates of $x\in \mathbb{R}^d$ by $x^{'}$. We …
- 351
10
votes
1
answer
472
views
A basic estimate of exponential sums
Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:
\begin{equation}
\sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\bigg|\leq C …
- 351