Expansion in the System of Shifts of a B-Spline

S. M. Lytkin

M. V. Lomonosov Moscow State University

Abstract: A method of approximating functions by linear combinations of shifts of a fixed function, a B-spline of the first, second, or third degree, is proposed. This method is exact for the classes of splines of the corresponding degree with deficiency 1. We obtain estimates of strong type for the norms of approximating operators in certain spaces, from which it follows immediately that the approximation in these spaces coincides in order with the best approximation by splines.

Keywords: spline, B-spline, periodic spline, system of shifts of a B-spline, deficiency of a spline, Hölder's inequality, Jensen's inequality.

UDC: 517.518

Received: 03.04.2008
Revised: 13.04.2009

DOI: 10.4213/mzm4738

 English version: Mathematical Notes, 2009, 86:6, 801–810

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