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 $(0, 1).$
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