All Questions
Tagged with fourier-transform harmonic-analysis
9 questions
25
votes
3
answers
13k
views
Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
- 31.5k
17
votes
2
answers
4k
views
Is this statement which relates the Fourier transform of a function to its singularities correct?
I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
- 698
3
votes
2
answers
1k
views
Composition of Riesz potentials
For $0<\alpha<n$ and $n\geq 2$ we define the Riesz potential by
$$
(I_\alpha f)(x) = \frac{1}{\gamma(\alpha)}
\int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, ,
\quad
\text{where}
\quad
\...
- 28k
26
votes
2
answers
3k
views
Image of L^1 under the Fourier Transform
The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
- 1,052
6
votes
1
answer
419
views
Positive-definiteness of radial sinc function in three dimensions
In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar ...
- 943
4
votes
1
answer
783
views
Fourier transform derivation from Laurent series
Using Laurent Series of a function $f(z)$ around a point $a \in \mathbb{C}$
$$f(z) = \sum^{\infty}_{n=-\infty} c_n(z-a)^n \ \ \ \ (1)$$
where
$$c_n = \frac{1} {2\pi i}\int\limits_{\gamma}\frac {f(z)} {...
- 51
3
votes
1
answer
518
views
Connection between the Fourier transform of f and |f|
If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and
$$
\|\widehat{f}\|_{L^{p'}}\...
- 425
2
votes
2
answers
916
views
Decay of the Fourier transform of a non-differentiable function
It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. ...
- 421
1
vote
0
answers
108
views
Recovering phase function using Fourier decomposition
I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function
$$f = e^{i \phi(x)}. $$
I am interested in the following problem. If I know the function/distribution $...
- 151