On Gelfand-Shilov spaces of type $W$
Spaces of infinitely differentiable functions on ${\mathbb R}^n$ (more general than Gelfand-Shilov spaces of type $W_M$) are considered in the article. Paley-Wiener type theorems are obtained.
math.FA↗
arXiv subjects
I. Kh. Musin
Publications and source records attributed to I. Kh. Musin.
On weighted polynomial approximation
Let $\varPhi:{\mathbb R}^n \to [1, \infty)$ be a semi-continuous from below function such that $\lim \limits_{x \to \infty} \displaystyle \frac {\ln \varPhi(x)} {\Vert x \Vert} = +\infty$. It is shown that polynomials are dense in $C_{\varPhi}({\mathbb R}^n)$.
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On a class of infinitely differentiable functions in ${\mathbb R}^n$ admitting holomorphic extension in ${\mathbb C}^n$
A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{φ_m\}_{m=1}^{\infty}$ of real-valued functions $φ_m \in~C({\mathbb R}^n)$ and a logarithmically convex sequence $M$ of positive numbers is considered in the article. In view of conditions on $M$ each function of $G(M, \varPhi)$ can be extended to an entire function in ${\mathbb C}^n$. Imposed conditions on $M$ and $\varPhi$ allow to describe the space of such extensions.
math.CV↗
On a Hilbert space of entire functions
A weighted Hilbert space $F^2_φ$ of entire functions of $n$ variables is considered in the paper. The weight function $φ$ is a convex function on ${\mathbb C}^n$ depending on modules of variables and growing at infinity faster than $a \Vert z \Vert$ for each $a > 0$. The problem of description of the strong dual of this space in terms of the Laplace transformation of functionals is studied in the article. Under some additional conditions on $φ$ the space of the Laplace transforms of linear continuous functionals on $F^2_φ$ is described. The proof of the main result is based on new properties of the Young-Fenchel transformation and a result of R.A. Bashmakov, K.P. Isaev and R.S. Yulmukhametov on asymptotics of multidimensional Laplace transform.
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On a space of entire functions and its Fourier transform
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on ${\mathbb R}^n$ and a Paley-Wiener type theorem are obtained.
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On perturbation of a surjective convolution operator
Let $μ\in {\cal E}'({\mathbb R}^n)$ be a compactly supported distribution such that its support is a convex set with non-empty interior. Let $X_2$ be a convex domain in ${\mathbb R}^n$, $X_1 = X_2 + supp \ μ$. Assuming that a convolution operator $A: {\cal E}(X_1) \to {\cal E}(X_2)$ acting by the rule $(Af)(x) = (μ* f)(x)$ is surjective we provide a condition on a linear continuous operator $B: {\cal E}(X_1) \to {\cal E}(X_2)$ that guarantees surjectivity of the operator $A+B$.
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On a space of entire functions and its Fourier transformation
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions (not radial in general) is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb R}^n$ and Paley-Wiener type theorem are given.
math.CV↗
On a space of entire functions rapidly decreasing on ${\mathbb R}^n$ and its Fourier transformation
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb R}^n$ and Paley-Wiener type theorem are obtained.
math.CV↗
On a space of rapidly decreasing infinitely differentiable functions on an unbounded convex set in ${\mathbb R}^n$ and its dual
Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.
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On a Space of Infinitely Differentiable Functions on an Unbounded Convex Set in ${\mathbb R}^n$ Admitting Holomorphic Extension in ${\mathbb C}^n$ and its Dual
We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
math.CV↗
Approximation by polynomials in a weighted space of infinitely differentiable functions
The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result for description of strong dual for weighted spaces of infinitely differentiable functions on real line and weighted spaces of sequences of infinitely differentiable functions on real line in terms of the Fourier-Laplace transform of functionals.
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Representation of Infinitely Differentiable Functions by Dirichlet Series
The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.
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On the Fourier-Laplace Transform of Functionals on a Weighted Space of Infinitely Differentiable Functions
The strong dual space of linear continuous functionals on a weighted space G of infinitely differentiable functions defined on the real line is described in terms of their Fourier-Laplace transforms.
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