Restoration of functions, integrals, and solutions to the heat conductivity equation from the Ulyanov $U_2$-classes

The paper dealt with a problem of numerical integration, and approximate restoration of functions and solutions to the heat conductivity equation with functions of distribution of starting temperatures from the classes $U_2(\beta,\theta,\alpha)$ defined by the rate of decreasing the trigonometric Fourier coefficients. Optimal orders of errors of the quadrature formulas, restoration, and discretization by the trigonometric Fourier coefficients in $L_2$ and $L_{\infty}$ metrics are obtained.