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 $|x|\to \infty$.

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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^{...
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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 $-\infty<x<\infty$ , $k,q=0,1,2,...$...
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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 ...
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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\...
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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 $C_{q}$ ...
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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 $...