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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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
$$\frac{|h|^{q}}{q!}C_{0}B^{q}q^{q}=\frac{1}{B^{q}e^{q}}\frac{1}{q^{q+{1/2}}e^{-q}\sqrt{2\...
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Minimum estimation
Hi could anyone explain me how one come sup with the term circled with red? The proof tries to show that for integer numbers the mimimum is somewhere else (a bit higher) than it would be if f would be ...
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Question about $|x^{k}\varphi^{(q)}(x)| \leqslant C_{q}A^{k}k^{k\alpha}$ in Gelfand Shilov spaces
I am reading into Gelfand Shilov spaces and in the book about distribution theory there is this space $S_{\alpha}$ defined by
$$|x^{k}\varphi^{(q)}(x)| \leqslant C_{q}A^{k}k^{k\alpha}$$
where $C_{q}$ ...
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Is there any compactly supported function in the Gelfand-Shilov space $\Sigma_1(=S^{(1)})$?
Good evening, I have a question related to the Gelfand-Shilov spaces as defined in https://arxiv.org/pdf/1505.04096.pdf (see 1.2, pag.2).
I seemed to understand that there is no non-zero function of $...
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1answer
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Doubts related to Differential Operator of Infinite order
Let $$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 ...
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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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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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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^{...
