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 517.5In the space $L_2[(0,1);x],$ by using a system of functions $\left\{ \widehat{J}_{ u}(μ_{k, u} x) \right\}_{k \in \mathbb{N}},$ $ u \geqslant 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 $m$th order $Δ^m_{γ(h)}(f),$ $m \in \mathbb{N},$ $h \in (0,1),$ and the generalized characteristics of smoothness $Φ^{(γ)}_{m}(f,t)= (1/t) \displaystyle\intt̂_0\|Δ^m_{γ(τ)}(f)\| d τ.$ For the classes $\mathcal{W}^{r, u}_2(Φ^{(γ)}_{m}, Ψ)$ 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 $(0, 1).$