Convergence of the Lagrange–Sturm–Liouville processes for continuous functions of bounded variation

The uniform convergence within an interval $(a,b)\subset [0,\pi]$ of Lagrange processes in eigenfunctions $L_n^{SL}(f,x)=\sum\nolimits_{k=1}^{n}f(x_{k,n})\frac{U_n(x)}{U_{n}'(x_{k,n})(x-x_{k,n})}$ of the Sturm–Liouville problem is established. (Here $0<x_{1,n}<x_{2,n}<\dots<x_{n,n}<\pi$ denote the zeros of the eigenfunction $U_n$ of the Sturm–Liouville problem.) A continuous functions $f$ on $[0,\pi]$ which is of bounded variation on $(a,b)\subset [0,\pi]$ can be uniformly approximated within the interval $(a,b)\subset [0,\pi]$. A criterion for uniform convergence within an interval $(a,b)$ of the constructed interpolation processes is obtained in terms of the maximum of the sum of the moduli of divided differences of the function $f$. Outside the interval $(a, b)$, the Lagrange interpolation process may diverge. The boundedness in the totality of the Lagrange fundamental functions constructed from eigenfunctions of the Sturm–Liouville problem is established. The case of the regular Sturm–Liouville problem with a continuous potential of bounded variation is also considered. The boundary conditions for the third kind Sturm–Liouville problem without Dirichlet conditions are studied. In the presence of service functions for calculating the eigenfunctions of the regular Sturm–Liouville problem, the Lagrange–Sturm–Liouville operator under study is easily implemented by computer technology.