
This article is cited in 2 scientific papers (total in 2 papers)
Meansquare approximation of functions by Fourier–Bessel series and the values of widths for some functional classes
K.Tukhliev^{} ^{} Khujand State University
Abstract:
It is known that many of the problems of mathematical physics,
reduced to a differential equation with partial derivatives written
in cylindrical and spherical coordinates, by using method of
separation of variables, in particular, leads to the Bessel
differential equation and Bessel functions. In practice, especially
in problems of electrodynamics, celestial mechanics and modern
applied mathematics most commonly used Fourier series in orthogonal
systems of special functions. Given this, it is required to
determine the conditions of expansion of functions in series into
these special functions, forming in a given interval a complete
orthogonal system.
The work is devoted to obtaining accurate estimates of convergence
rate of Fourier series by Bessel system of functions for some
classes of functions in a Hilbert space $L_{2}:=L_{2}([0,1],x dx)$ of
square summable functions $f: [0,1]\rightarrow\mathbb{R}$ with the
weight $x.$
The exact inequalities of Jackson–Stechkin type on the sets of
$L_{2}^{(r)}(\mathcal{D}),$ linking $E_{n1}(f)_{2}$ — the best
approximation of function $f$ by partial sums of order $n1$ of the
Fourier–Bessel series with the averaged positive weight of
generalized modulus of continuity of $m$ order $\Omega_{m}(\mathcal{D}^{r}f; t),$ where $\mathcal{D}:=\frac{d^{2}}{dx^{2}}+\frac{1}{x}\cdot\frac{d}{dx}
\frac{\nu^{2}}{x^{2}}$ — is a Bessel differential operator of
secondorder of first kind index $\nu$. Similar inequalities are
also obtained through the $\mathcal{K}$functionals $r$s derivatives
of functions.
The exact value of the different $n$widths for classes of functions
defined by specified characteristics, in $L_{2}$ were calculated.
Keywords:
Bessel function, best approximation, $\mathcal{K}$functional, generalized modulus of continuity of $n$th order, Fourier–Bessel series, $n$widths.
DOI:
https://doi.org/10.22405/222683832016174141156
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UDC:
517.5 Received: 12.09.2016 Accepted:12.12.2016
Citation:
K.Tukhliev, “Meansquare approximation of functions by Fourier–Bessel series and the values of widths for some functional classes”, Chebyshevskii Sb., 17:4 (2016), 141–156
Citation in format AMSBIB
\Bibitem{Tuk16}
\by K.Tukhliev
\paper Meansquare approximation of functions by FourierBessel series and the values of widths for some functional classes
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 4
\pages 141156
\mathnet{http://mi.mathnet.ru/cheb522}
\crossref{https://doi.org/10.22405/222683832016174141156}
\elib{https://elibrary.ru/item.asp?id=27708211}
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This publication is cited in the following articles:

Mukim S. Saidusajnov, “$\mathcal{K}$functionals and exact values of $n$widths in the Bergman space”, Ural Math. J., 3:2 (2017), 74–81

M. Sh. Shabozov, M. S. Saidusajnov, “Upper Bounds for the Approximation of Certain Classes of Functions of a Complex Variable by Fourier Series in the Space $L_2$ and $n$Widths”, Math. Notes, 103:4 (2018), 656–668

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