Approximation in the mean of the classes of functions in the space $L_2[(0,1);x]$ by the Fourier–Bessel sums and estimates of the values of their $n$-widths
Abstract
UDC 517.5
In the space $L_2[(0,1);x],$ by using a system of functions $\left\{ \widehat{J}_{\nu}(\mu_{k,\nu} x) \right\}_{k \in \mathbb{N}},$ $\nu \geqslant 0,$ orthonormal with weight $x$ and formed by the Bessel function of the first kind of index $\nu$ and its positive roots, we construct the generalized finite differences of the $m$th order $\Delta^m_{\gamma(h)}(f),$ $m \in \mathbb{N},$ $h \in (0,1),$ and the generalized characteristics of smoothness $\Phi^{(\gamma)}_{m}(f,t)= (1/t) \displaystyle\int\^t_0\|\Delta^m_{\gamma(\tau)}(f)\| d \tau.$ For the classes $\mathcal{W}^{r,\nu}_2(\Phi^{(\gamma)}_{m}, \Psi)$ defined by using the differential operator $D^r_\nu,$ the function $\Phi^{(\gamma)}_{m}(f),$ and the majorant $\Psi,$ we establish estimates from the lower and upper of the values of a series of $n$-widths. A condition for $\Psi,$ which allows us to compute the exact values of $n$-widths is established. To illustrate our exact results, we present several specific examples. We also consider the problems of absolute and uniform convergence of Fourier–Bessel series on the interval $(0, 1).$
References
В. С. Владимиров, Уравнения математической физики, 3-е изд., Наука, Москва (1976).
И. К. Даугавет, Введение в теорию приближений, Ленинград (1977).
Д. Гайер, Лекции по теории аппроксимации в комплексной плоскости, Мир, Москва (1986).
S. B. Vakarchuk, On the estimates of widths of the classes of continuty and majorants in the weighted space $L_{2,x}(0,1)$, Ukr. Math. J., 71, № 2, 202–214 (2019).
J. Boman, H. S. Shapiro, Comparison theorems for a generalized modulus of continuity, Ark. Mat., 9, № 1, 91–116 (1971).
J. Boman, Equivalence of generalized moduli of continuity, Ark. Mat., 18, № 1, 73–100 (1980).
Z. Ditzian, V. Totik, Moduli of smoothness, Springer Ser. Comput. Math., 9, Springer-Verlag, New York (1987).
Б. Сендов, В. Попов, Усредненные модули гладкости, Мир, Москва (1988).
R. M. Trigub, Absolute convergence of Fourier integrals, summability of Fourier series and polynomial approximation of functions on the torus, Izv. Math., 17, № 3, 567–593 (1981).
K. G. Ivanov, On a new characteristic of functions. I., Сердика Бълг. Мат. Списание, 8, № 3, 262–279 (1982).
K. G. Ivanov, On a new characteristic of functions. II. Direct and converse theorems for the best algebraic approximation in $C[-1, 1]$ and $L_p[-1, 1]$, Плиска Бълг. Мат. Студ., 5, 151–163 (1983).
К. М. Потапов, Аппроксимация многочленами на конечном отрезке вещественной оси, Тр. междунар. конф. по конструктивной теории функций, Варна, 1–5 июня 1981 г., Болг. АН, София (1983), с. 134–143.
K. V. Runovskii, On approximation by families of linear polynomial operators in $L_p$-space, $0<р<1$, Sb. Math., 82, № 2, 441–459 (1995).
N. N. Pustovoitov, Estimates of the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences and the multidimensional Jackson's theorem, Sb. Math., 188, № 10, 1507–1520 (1997).
А. Г. Бабенко, О неравенстве Джексона–Стечкина для наилучших $L^2$-приближений функций тригонометрическими полиномами, Тр. Института математики и механики УрО РАН, 6, 1–19 (1999).
V. A. Abilov, F. V. Abilova, Approximation of functions by algebraic polynomials in the mean, Russian Math., 41, № 3, 60–62 (1997).
С. Н. Васильев, Точное неравенство Джексона–Стечкина в $L_2$ с модулем непрерывности, порожденным произвольным конечно-разностным оператором с постоянными коэффициентами, Докл. АН, 385, № 1, 11–14 (2002).
A. I. Kozko, A. V. Rozhdestvenskii, On Jackson's inequality for generalized moduli of continuity, Math. Notes, 73, № 5, 736–741 (2003).
V. A. Abilov, F. V. Abilova, Problems in the approximation of $2pi$-periodic functions by Fourier sums in the space $L_2$, Math. Notes, 76, № 6, 749–757 (2004).
S. B. Vakarchuk, On best polynomial approximations in $L_2$, Math. Notes, 70, № 3, 300–310 (2001).
S. B. Vakarchuk, Exact constants in Jackson-type inequalities and exact values of the widths of function classes in $L_2$, Math. Notes, 78, № 5-6, 735–739 (2005).
S. B. Vakarchuk, V. I. Zabutnaya, A sharp inequality of Jackson–Stechkin type in $L_2$ and the widths of functional classes, Math. Notes, 86, № 3, 306–313 (2009).
S. B. Vakarchuk, V. I. Zabutnaya, Jackson–Stechkin type inequalities for special moduli of continuity and widths of function classes in the space $L_2$, Math. Notes, 92, № 4, 458–472 (2012).
S. B. Vakarchuk, The best mean square approximation of functions, given at the real axis by entire functions of exponential type, Ukr. Math. J., 64, № 5, 754–767 (2012).
S. B. Vakarchuk, M. Sh. Shabozov, V. I. Zabutnaya, Structural characteristics of functions from $L_2$ and the exact values of widths of some functional classes, J. Math. Sci., 206, № 1, 97–114 (2015).
V. A. Abilov, F. V. Abilova, V. R. Kerimov, Sharp estimates for the convergence rate of Fourier–Bessel series, Comput. Math. and Math. Phys., 55, № 7, 1094–1102 (2015).
S. B. Vakarchuk, Generalized smoothness characteristics in Jackson-type inequalities and widths of classes of functions in $L_2$, Math. Notes, 98, № 4, 572–588 (2015).
S. B. Vakarchuk, V. I. Zabutnaya, Inequalities between best polynomial approximations and some smoothness characteristics in the space $L_2$ and widths of classes of functions, Math. Notes, 99, № 2, 222–242 (2016).
S. B. Vakarchuk, Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths for the classes of $(psi,beta)$-differentiable functions in $L_2. $ I, Ukr. Math. J., 68, № 6, 823–848 (2016).
S. B. Vakarchuk, Jackson-type inequalities with generalized
modulus of continuity and exact values of the $n$-widths for the classes of $(psi,beta)$-differentiable functions in $L_2. $ III, Ukr. Math. J., 68, № 10, 1299–1319 (2017).
К. Тухлиев, Среднеквадратическое приближение функций рядами Фурье–Бесселя и значения поперечников некоторых функциональных классов, Чебышев. сб.,17, № 4, 141–156 (2016).
S. B. Vakarchuk, Widths of some classes of functions defined by the generalized moduli of continuity $omega_{gamma}$ in the space $L_2$, J. Math. Sci., 227, № 1, 105–115 (2017).
S. B. Vakarchuk, Generalized characteristics of smoothness and some extreme problems of the approximation theory of functions in the space $L_2(mathbb{R}). $ I, Ukr. Math. J., 70, № 9, 1345–1374 (2019).
S. B. Vakarchuk, Generalized characteristics of smoothness and some extreme problems of the approximation theory of functions in the space $L_2(mathbb{R}). $ II, Ukr. Math. J., 70, № 10, 1550–1584 (2019).
S. B. Vakarchuk, Best polynomial approximations and widths of classes of functions in the space $L_2$, Math. Notes, 103, № 2, 308–312 (2018).
S. B. Vakarchuk, On estimates in $L_2(mathbb{R})$ of mean $nu$-widths of classes of functions defined via the generalized modulus of continuity of $omega_{M}$, Math. Notes, 106, № 2, 191–202 (2019).
F. Abdullayev, S. Chaichenko, A. Shidlich, Direct and inverse approximation theorems of functions in the Musielak–Orlicz type space, Math. Inequal. and Approx., 24, № 4, 323–336 (2021).
F. Abdullayev, S. Chaichenko, M. Imashysy, A. Shidlich, Jackson-type inequalities and widths of functional classes in the
Musielak–Orlicz type space, Rocky Mountain J. Math., 51, № 4, 1143–1155 (2021).
S. O. Chaichenko, A. L. Shidlich, T. V. Shulyk, Direct and inverse approximation theorems in the Besicovitch–Musielak–Orlicz spaces of almost periodic functions, Ukr. Mat. Zh., 74, № 5, 701–716 (2022).
Б. М. Макаров, Л. В. Флоринская, Теория меры и интеграла., Вып. 1. Мера. Измеримые функции, Изд-во Ленинград. ун-та, Ленинград (1974).
И. Г. Арамович, Р. С. Гутер, Л. А. Люстерник, М. И. Раухваргер, М. И. Сканави, А. Р. Янпольский, Математический анализ (дифференцирование и интегрирование), Физматгиз, Москва (1961).
Б. Г. Коренев, Введение в теорию бесселевых функций, Наука, Москва (1971).
Г. П. Толстов, Ряды Фурье, 3-е изд., Наука, Москва (1980).
Н. Г. де Брейн, Асимптотические методы в анализе, Изд-во иностр. лит., Москва (1961).
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