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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.
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
3
votes
Is there a function that is not absolutely integrable in [−π,π] so that its Fourier Series E...
All $2π$ periodic distributions are temperate and since all temperate distributions have a Fourier transform, you have plenty of examples. Note that the previous statements prove the power of abstract …
- 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
2
votes
convolution algebra on a compact surface in $\mathbb{R}^3$
The Schwartz space itself, $\mathscr S(\mathbb R^n)$, can be viewed as a subspace of smooth functions defined on the sphere $\mathbb S^n$, the unit sphere of $\mathbb R^{n+1}.$ We recall that
$$
\math …
- 16.2k
1
vote
Accepted
A question which belongs to a class of Zygmund functions
The standard Zygmund class (with $\epsilon=0$) is the Besov space
$
B^1_{\infty,\infty},
$
that is using a Littlewood-Paley decomposition
$$
1=\sum_{\nu\ge 0}\varphi_\nu(\xi),\quad \varphi_\nu(\xi) =\ …
- 16.2k
3
votes
Accepted
Asymptotics of a one-parameter family of Schwartz functions
$$
I(\tau)=\int_{\mathbb R}e^{-\pi\frac{\tau}{\pi} x^2}e^{-2i\pi \tau x (1-\frac{1}{2i\pi})}dx=
(\frac{\pi}{\tau})^{1/2}e^{-\pi\frac{\pi}{\tau} \tau^2 (1-\frac{1}{2i\pi})^2}=
(\frac{\pi}{\tau})^{1/2}e …
- 16.2k
3
votes
Accepted
Fourier transform of a differential operator
Let me change slightly your notations and consider the quadratic form in $\mathbb R_{\xi,\eta}^2$
$$
Q(\xi,\eta)=\alpha \xi ^2+2\gamma \xi \eta+\beta \eta^2,
$$
where $\alpha, \beta$ are real paramete …
- 16.2k
3
votes
Accepted
Decay of oscillatory integral for non-analytic phase function
Let me fix $\delta=1$ for simplicity. Let us use a Van der Corput method. We have for $\epsilon\in (0,1)$ to be chosen later, with $\phi(x)= e^{-x^{-2}}, $ noting that $\phi'(x)=\phi(x) 2 x^{-3}$
$$
I …
- 16.2k
0
votes
0
answers
119
views
Wigner distribution
The Wigner distribution of $u\in L^2(\mathbb R)$ is defined as a function $W(u)$ on $\mathbb R^2$ given by
$$
W(u)(x,\xi)=\int_\mathbb R u\left(x+\tfrac z2\right) \overline{u\left(x-\tfrac z2\right)} …
- 16.2k
3
votes
Accepted
Fourier analysis and fractional calculus
Too long for a comment. Your spelling of the name of the mathematician Joseph Fourier is incorrect. Also your formula is almost impossible to read: your fractional derivative on the lhs is the Fourier …
- 16.2k
1
vote
Logarithm of the Fourier transform?
Let us start with the identity $F^4=I$. As a result, we have formally
\begin{align}
\ln F&=\ln(I+F-I)=\sum_{k\ge 1}\frac{(-1)^{k-1}(F-I)^k}{k}
\\&=
\sum_{1\le k\le 3}\frac{(-1)^{k-1}(F-I)^k}{k}+
\sum_ …
- 16.2k
2
votes
0
answers
274
views
Hilbert transform
Let $\mathscr H$ be the Hilbert transform, that is the Fourier multiplier by $\text{sign } \xi$:
$$
(\mathscr H u)(x)=\int_{\mathbb R} e^{2iπ x\xi }(\text{sign } \xi) \hat u(\xi) d\xi.
$$
The Hilbert …
- 16.2k
1
vote
Relation between inverse Fourier transform of dyadic bump function and its regularity
I suggest a mild modification of your question. In the first place, let us set
$
\psi_k(t)=\psi_1(t2^{-k}).
$
Let us also replace $\sqrt{1+\vert \xi\vert^2}$ by $\vert \xi\vert$. We get
$$
\int_{\math …
- 16.2k
1
vote
Inverse Fourier transform of an $L^2$ function as limit on balls
Let me reformulate your question: you consider the Fourier multiplier
$
L_m=\mathbf 1_{B_m}(D)
$
defined by the formula
$$
(\mathbf 1_{B_m}(D)f)(x)=\int e^{2iπ x\xi}\mathbf 1_{B_m}(\xi) \hat f(\xi) d\ …
- 16.2k
2
votes
0
answers
125
views
Fourier multiplier on $L^1$
On the Wikipedia page,
one can read that an iff condition for L1 boundedness of the Fourier multiplier m(D) is that
$$
\hat m\quad\text{ is a Borel measure with finite total mass. }
$$
There is no ref …
- 16.2k