Best quadrature formulas and methods of reconstructing functions defined by variation diminishing kernels

The author considers classes of periodic functions given by variation diminishing convolutions. These classes include, for example, the Sobolev class $W_p^r(\mathbf T)$ and the class $K_p^U(\mathbf T)=\{f|\|U(d/dx)f(\cdot)\|_p\leqslant1\}$, where $U$ is any polynomial with real coefficients and real roots. The Poisson kernel, the de la Vallée–Poussin kernel and many others have the variation diminishing property.
For classes given by variation diminishing convolutions the author proves optimality of the rectangle formula among all quadrature formulas of the form $\sum^k_{i=1}\sum^{\nu_i-1}_{j=0}a_{ij}f^{(j)}(x_i)$ with $\sum_{i=1}^k\nu_i\leqslant N$. In addition, a solution of Favard's problem is given and an optimal method of reconstructing functions of these classes is found.
Bibliography: 22 titles.