Approximation in the Mean for the Classes Of Functions in the Space L2[(0, 1); x] by The Fourier–Bessel Sums And Estimation of the Values of Their n-Widths

Abstract

In the space L₂[(0, 1); x], by using a system of functions \({\left\{{\widehat{J}}_{v}\left({\mu }_{k,v}x\right)\right\}}_{k\in {\mathbb{N}}}, v\ge 0,\) orthonormal with weight x and formed by a Bessel function of the first kind of index v and its positive roots, we construct generalized finite differences of the mth order \({\Delta }_{\gamma \left(h\right)}^{m}\left(f\right),\) m ∈ ℕ, h ∈ (0, 1), and the generalized characteristics of smoothness \({\Phi }_{\gamma \left(h\right)}^{\left(\gamma \right)}\left(f,t\right)=\left(1/t\right)\underset{0}{\overset{t}{\int }}\Vert {\Delta }_{\gamma \left(\tau \right)}^{m}\left(f\right)\Vert d\tau .\) For the classes \({\mathcal{W}}_{2}^{r,v}{\Phi }_{m}^{\left(\gamma \right)},\left(\uppsi \right)\) defined by using the differential operator \({D}_{v}^{r},\) the function \({\Phi }_{m}^{\left(\gamma \right)}\left(f\right),\) and the majorant ψ, we establish lower and upper estimates for the values of a series of n-widths. We established the condition for ψ, which enables us to compute the exact values of n-widths. 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).