On a theorem of Titchmarch-Kodaira-Weidmann concerning absolutely continuous operators, I

For the Favard class F_r in the space C_2π of continuous 2π-periodic functions we solve the following problem. Given x∈$\text{R}$ and knots x₀< x₁ < ··· < x_v?1., x_u? 2π we determine weights x_ki(0 ?k · n, 0 ? j < r) such that

$\text{sup}\text{f∈F}{}_{\mathrm{r}}\text{f(x)?}\text{∑}\text{k=0}\text{n?1}\text{∑}\text{j=0}\text{r?1}\text{α}{}_{\mathrm{kj}}\text{f}{}^{\mathrm{(i)}}\text{(X}{}_{\mathrm{k}}\text{)}$

is minimal. The optimal weights are unique (except for a trivial case) and we obtain them from a system of periodic polynomial splines u_kj(0 ? k < n, 0 ?j< r): α_kj = u_kj(x). These splines induce an interpolation operator whose degree of approximation with respect to the class F_r is minimal if the knots are equidistant. Finally, we describe an efficient numerical procedure which shows how to compute the interpolation spline in the equidistant case.