Questions tagged [gelfand-shilov-spaces]

The Gelfand Shilov Spaces are spaces of fundamental functions. Conditions are imposed on both the decrease of fundamental functions as well as the growth of their derivatives as $|x|\to \infty$.

Filter by
Sorted by
Tagged with
0
votes
0answers
14 views

Behavior of convergence of a certain remainder term

I would have a question to this part of a proof. If I set $|h|=1/Be$ I am getting the following calculation $$\frac{|h|^{q}}{q!}C_{0}B^{q}q^{q}=\frac{1}{B^{q}e^{q}}\frac{1}{q^{q+{1/2}}e^{-q}\sqrt{2\...
1
vote
1answer
23 views

Minimum estimation

Hi could anyone explain me how one come sup with the term circled with red? The proof tries to show that for integer numbers the mimimum is somewhere else (a bit higher) than it would be if f would be ...
0
votes
0answers
14 views

Question about $|x^{k}\varphi^{(q)}(x)| \leqslant C_{q}A^{k}k^{k\alpha}$ in Gelfand Shilov spaces

I am reading into Gelfand Shilov spaces and in the book about distribution theory there is this space $S_{\alpha}$ defined by $$|x^{k}\varphi^{(q)}(x)| \leqslant C_{q}A^{k}k^{k\alpha}$$ where $C_{q}$ ...
0
votes
0answers
12 views

Is there any compactly supported function in the Gelfand-Shilov space $\Sigma_1(=S^{(1)})$?

Good evening, I have a question related to the Gelfand-Shilov spaces as defined in https://arxiv.org/pdf/1505.04096.pdf (see 1.2, pag.2). I seemed to understand that there is no non-zero function of $...
2
votes
1answer
190 views

Doubts related to Differential Operator of Infinite order

Let $$f(s)=\sum_0^\infty c_vs^v$$ be some entire function. We say that the differential operator $f(d/dx)=\sum_0^\infty c_vd^v/dx^v$ is defined in some fundamental space $\varPhi$, if for any $\varphi ...
3
votes
0answers
47 views

Doubt about inequalities

My original question is if $f\in \mathcal{S}_{\alpha_1}^{\beta_1},\: g \in \mathcal{S}_{\alpha_2}^{\beta_2}$, where does $(f\cdot g) (x)=f(x)g(x)$ belong ? where $\mathcal{S}_\alpha^\beta$ is defined ...
5
votes
0answers
254 views

Laplace transform of functions related to type $\mathcal{S}$, and the relation to entire functions

I have doubts in the following two questions : What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , $k,q=0,1,2,...$...
12
votes
0answers
465 views

How to construct examples of functions in the Spaces of type $\mathcal{S}$

There are $3$ $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. They are defined by: $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{...