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 - Vakarchuk, S.; Vakarchuk, M. (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 517.5 In the space L_2[(0,1);x], by using a system of functions { J_ u(μ_k, u x) }_k N , u 0, orthonormal with weight x and formed by the Bessel function of the first kind of index u and its positive roots, we construct the generalized finite differences of the mth order Δ^m_γ(h)(f), m N, h (0,1), and the generalized characteristics of smoothness Φ^(γ)_m(f,t)= (1/t) ̂_0\|Δ^m_γ(τ)(f)\| d τ. For the classes W^r, u_2(Φ^(γ)_m, Ψ) defined by using the differential operator D^r_ u, the function Φ^(γ)_m(f), and the majorant Ψ, we establish estimates from the lower and upper of the values of a series of n-widths. A condition for Ψ, 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).