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Results tagged with fourier-analysis
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user 21907
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.
18
votes
3
answers
7k
views
Eigenvectors of the Fourier transformation
The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$
by
$
\hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx.
$
It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the …
- 16.2k
11
votes
Accepted
Fourier Coefficients and Hölder Continuity
There is an excellent characterization of Hölder spaces via the Fourier transform, using Besov spaces. Let $\alpha\in (0,1)$: a function $u$
defined on $\mathbb R^n$ belongs to $L^\infty\cap C^\alpha$ …
- 16.2k
9
votes
Accepted
Fourier transform in $L^1$?
To fix notations let us consider the Fourier transformation as acting on functions over $\mathbb R^n$. The set of functions with Fourier transform $\hat{f} \in L^1(\mathbb{R}^n)$ is the Wiener algebra …
- 16.2k
7
votes
2
answers
467
views
Eigenstates of Fourier transformation
Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by
$$
(\mathcal F u)(\xi)=\int e^{-2iπ x\cd …
- 16.2k
6
votes
Are Besov spaces $B^{s}_{p,q}$ invariant under Fourier transform?
Let me note $\phi_k(D)$ the Fourier multiplier $\phi_k(\xi)$, i.e.
$
\text{Fourier}\bigl(\phi_k(D)u\bigr)(\xi)=\phi_k(\xi)\hat u(\xi).
$
$\bullet$ The answer to (1) is yes since
$$
\Vert{u}\Vert_{L^1 …
- 16.2k
5
votes
Easy Garding Inequality
I understand that you are dealing with semi-classical symbols
$$
b(x,\xi, h)=a(x,h\xi), \quad \vert\partial_x^\alpha\partial_\xi^\beta b\vert\le
C_{\alpha\beta} h^{\vert \beta\vert} m(x),\quad 0<h\le …
- 16.2k
5
votes
Is there a compactly supported function that its Fourier transfrom vanishes at given n real ...
Consider the polynomial
$
P(\xi)=\prod_{1\le j\le n}(\xi-\lambda_j).
$
The inverse Fourier transform of $(\xi-\lambda_j)$ is
$$
\int(\xi-\lambda_j) e^{2iπ x\xi} d\xi=(D_x-\lambda_j)(\delta_0)=\frac{\ …
- 16.2k
5
votes
fourier analytic proofs
Let me speak about the "Triumph of Fourier" according to the words of Laurent Schwartz in his autobiography. The Fourier transformation is a handy tool to characterize regularity of functions.
Let $ …
5
votes
Accepted
About the boundedness of a multiplication operator.
When $p=2$, boundedness is a triviality and it is the only trivial case. It is not true for $p=1$ nor for $p=\infty$, although the Fourier multiplier $sign(D_x)$ sends $L^1$ into $L^1_w$ and the Marci …
- 16.2k
5
votes
Fourier transform of Analytic Functions
The following basic result needs to be quoted on these matters of analyticity: the Paley-Wiener-Schwartz theorem gives a characterization of distributions with compact support. Let $u$ be a tempered d …
- 16.2k
5
votes
Accepted
Sobolev convergence of Fourier series
Let us start with pointing out that $f\in H^\sigma$ is equivalent to
$$
(\langle n\rangle^\sigma\hat f(n))_{n\in \mathbb Z}\in \ell^2(\mathbb Z),
\quad \text{with $\langle n\rangle=\sqrt{1+n^2}$.}
$$ …
- 16.2k
4
votes
Accepted
Real-analytic variant of theorem 4.2.5 of Duistermaat's "FIO", 1996
You may be able to read the Sato-Kawai-Kashiwara lecture notes if your algebraic geometry background is sufficient for this non-trivial task.
On the other hand, the book by J. Sjöstrand "Singularité …
- 16.2k
4
votes
0
answers
925
views
Norms of the Dirichlet kernel
I guess that the following estimates are classical. Let $D_N$ be the $1D$ Dirichlet kernel,
$$
D_N(t)=\frac{\sin((N+\frac12)t)}{\sin (t/2)}.
$$
We have for $1<p<\infty$,
\begin{align}
\Vert D_N\Vert_{ …
- 16.2k
4
votes
Accepted
Stationary phase method for $\varphi''(x_0)= 0$
Let me assume that $a=-\infty, b=+\infty, x_0=0$ and $f$ smooth and compactly supported near 0. Then after a suitable change of variable, you get that
$
I(\lambda)=\int g(t) e^{i\lambda t^3/3} dt,
$
…
- 16.2k
4
votes
Accepted
Fourier Transform: Smoothness and Decay
Your space is some sort of Besov space,
containing $B^{2+\epsilon}_{1,\infty}$.
Using a Littlewood-Paley decomposition $1=\sum_\nu\varphi_\nu(\xi)$
with $\varphi_0\in C^\infty_c$, $\varphi_\nu(\xi) …
- 16.2k