Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


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Trynin A. Y., Kireeva E. D. The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange – Sturm – Liouville. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 1, pp. 51-63. DOI: 10.18500/1816-9791-2020-20-1-51-63, EDN: YRYAST

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Published online: 
02.03.2020
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Russian
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YRYAST

The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange – Sturm – Liouville

Autors: 
Trynin Aleksandr Yurievich, Saratov State University
Kireeva Ekaterina Dmitrievna, Saratov State University
Abstract: 

Let us say that the principle of localization holds at the class of functions F at point x0 ∈ [0, π] for the Lagrange –Sturm – Liouville interpolation process LSLn (f, x) if limn→∞ LSLn (fx0) − LSL(g, x0) = 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 Oδ(x0), δ > 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, π] 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 x0 ∈ [0, π]. in both cases depend solely on the values of the approximate function just in the neighborhood of this point x0 ∈ [0, π].

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Received: 
31.10.2018
Accepted: 
15.12.2018
Published: 
02.03.2020