Mean-square approximation of functions by Fourier–Bessel series and the values of widths for some functional classes

It is known that many of the problems of mathematical physics, reduced to a differential equation with partial derivatives written in cylindrical and spherical coordinates, by using method of separation of variables, in particular, leads to the Bessel differential equation and Bessel functions. In practice, especially in problems of electrodynamics, celestial mechanics and modern applied mathematics most commonly used Fourier series in orthogonal systems of special functions. Given this, it is required to determine the conditions of expansion of functions in series into these special functions, forming in a given interval a complete orthogonal system.
The work is devoted to obtaining accurate estimates of convergence rate of Fourier series by Bessel system of functions for some classes of functions in a Hilbert space $L_{2}:=L_{2}([0,1],x dx)$ of square summable functions $f: [0,1]\rightarrow\mathbb{R}$ with the weight $x.$
The exact inequalities of Jackson–Stechkin type on the sets of $L_{2}^{(r)}(\mathcal{D}),$ linking $E_{n-1}(f)_{2}$ — the best approximation of function $f$ by partial sums of order $n-1$ of the Fourier–Bessel series with the averaged positive weight of generalized modulus of continuity of $m$ order $\Omega_{m}(\mathcal{D}^{r}f; t),$ where $\mathcal{D}:=\frac{d^{2}}{dx^{2}}+\frac{1}{x}\cdot\frac{d}{dx}- \frac{\nu^{2}}{x^{2}}$ — is a Bessel differential operator of second-order of first kind index $\nu$. Similar inequalities are also obtained through the $\mathcal{K}$-functionals $r$-s derivatives of functions.
The exact value of the different $n$-widths for classes of functions defined by specified characteristics, in $L_{2}$ were calculated.