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DOI: 10.23671/VNC.2018.4.23390

# Convergence of the Lagrange-Sturm-Liouville Processes for Continuous Functions of Bounded Variation

Trynin, A. Y.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 4.
Abstract:
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.
Keywords: uniform convergence, sinc approximations, bounded variation, Lagrange-Sturm-Liouville processes.