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Results tagged with fourier-analysis
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user 75422
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.
1
vote
Composition operators on fractional-order (periodic) Sobolev spaces
Let $\frac12<s<1$, let $u\in H^s(\mathbb R)$, then also $u\in C_0$, and let $f$ be Lipschitz on $[-||u||_\infty \ ,+||u||_\infty]$ with $|f(x)-f(y)|\leq M|x-y|$. Then $$\int\int \frac{|f(u(t))-f(u(t') …
- 3,085
0
votes
Composition operators on fractional-order (periodic) Sobolev spaces
Hint at a partial answer to the revised question, extended to $p\in (0,\infty)$: for $\frac12<s<1$, $p\geq 1$ is necessary and sufficient.
The "if" part is straightforward using the double integral …
- 3,085
2
votes
Accepted
The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator
Here is an example where $T(f,g)\notin l^2$ while $m\in L^2$ and $f=g\in l^2$:
Let $a(n)=|n|^{-1+s}$ with $\frac38<s<\frac12$ (so that $a\in l^2$), $m(\xi,\eta)=\hat{a}(\xi)\hat{a}(\eta)$. Take also …
- 3,085