The principle of localization at the class of functions integrable in the Riemann for the processes of Lagrange–Sturm–Liouville

Let us say that the principle of localization holds at the class of functions $F$ at point $x_0 \in [0, \pi]$ for the Lagrange–Sturm–Liouville interpolation process $L_n^{SL}(f,x)$ if $\lim_{n \rightarrow \infty}\left|L_n^{SL}(f, x_0)-L_n^{SL}(g,x_0)\right|=0$ follows from the fact that the condition $f(x)=g(x)$ is met for any two functions f and g belonging to F in some neighborhood $O_\delta(x_0)$, $\delta>0$. It is proved that the principle of localization at the class of Riemann integrable functions holds for interpolation processes built on the eigenfunctions of the regular Sturm–Liouville problem with a continuous potential of bounded variation. It is established that the principle of localization at the class of continuous on the segment $[0, \pi]$ functions holds for interpolation processes built on the eigenfunctions of the regular Sturm–Liouville problem with an optional continuous potential of bounded variation. We consider the case of boundary conditions of the third kind, from which the boundary conditions of the first kind are removed. Approximative properties of Lagrange–Sturm–Liouville operators at point $x_0\in [0, \pi]$ in both cases depend solely on the values of the approximate function just in the neighborhood of this point $x_0\in [0, \pi]$.