3
votes
0answers
104 views

Fourier transform and support of a distribution

Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on ...
2
votes
1answer
147 views

Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwarz-Bruhat space ...
2
votes
0answers
170 views

Spectral gap of tempered distributions

Hi, Let $\Lambda\subset\mathbb{R}$ be an infinite discrete set of finite density (for simplicity one may take the density equals 1) and $\delta_{\lambda}$ is a unit mass located at the point ...
3
votes
1answer
364 views

Decomposition of distributions

Can we write every (tempered) distribution $\psi$, say on $\mathbb{R}$, as the sum of two distributions $\psi = \psi_1 + \psi_2$ such that $\psi_1$ and the Fourier transform of $\psi_2$ are ...
0
votes
1answer
399 views

($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$

Note: I first posted question on math.stackexchange and I got one reply, which was a bit helpful (I'm still trying to understand it fully), but did not explore the two solution cases that I mentioned. ...
0
votes
2answers
606 views

Inverse Fourier transform of class of infinitely differentiable function with compact support

For which $f \in S(R^n)$, the Schwartz class, $\hat f \in D(R^n)$ ?
1
vote
5answers
2k views

fourier transform of (real) exponential

Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?