Approximation in the mean for the classes of functions in the space \(L_2[(0, 1); x]\) by the Fourier-Bessel sums and estimation of the values of their \(n\)-widths (Q6597869)

Let \(L_2[(0,1);x]\) be the Hilbert space of functions \(f:(0,1)\rightarrow \mathbb{R}\) such that \(\sqrt{x}f(x)\) is square summable on the interval \((0,1)\), with the inner product \((f,g)=\int_0^1xf(x)g(x)dx\). With the help of Bessel function \(J_\nu\) of the first kind of index \(\nu (\nu \leq 0)\) and its positive roots \(\mu_{k\nu}, k\in\mathbb{N}\) a complete and orthonormal with weight \(x\) system of functions \(\hat{J_\nu}(\mu_{k\nu}x)\) is considered. The problems of absolute and uniform convergence of Fourier-Bessel series with respect to the system \(\hat{J_\nu}(\mu_{k\nu}x)\) and also the problem of approximation by partial sums of this series are studied. As a characteristic of smoothness of the functions \(f\in L_2[(0,1);x]\), the average norm of its generalized finite difference of the \(m\)-th order is considered. For classes, defined by using of some differential operator, characteristic of smoothness, and the so-called majorant function \(\Psi\), lower and upper estimates for the values of a series of \(n\)-widths are established. It is noteworthy that under some conditions on \(\Psi\) the exact values of \(n\)-widths are computed.