On end conditions for integro quadratic spline interpolation in the mean

Yu. S. Volkov^a, T. Zhanlav^b, R.-O. Mijiddorj<sup>c

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^b Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences
^c Mongolian National University of Education

Abstract: The problem of function recovery is investigated, where instead of function values at certain points of the segment, integrally averaged values over intervals are known. Yu. N. Subbotin proposed to call such a problem as interpolation in the mean. Using the relationship between integro quadratic splines that solve the specified problem and interpolation cubic splines, the issue of choosing end conditions is considered in the case of the absence of additional information at the ends of the segment that could be used as end conditions. When constructing an interpolation in the mean spline through B-splines, formulas are proposed for explicitly specifying the expansion coefficients at the ends of the segment, which ensure the preservation of the highest third order of approximation.

Keywords: integro spline, interpolation in the mean, B-splines, end conditions, cubic splines.

UDC: 519.65

MSC: 41A15

Received: 30.04.2025
Revised: 26.09.2025
Accepted: 06.10.2025

DOI: 10.21538/0134-4889-2025-31-4-95-105

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