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 Izv. Saratov Univ. Math. Mech. Inform., 2018, Volume 18, Issue 2, Pages 172–182 (Mi isu753)

Scientific Part
Mathematics

On binary B-splines of second order

S. F. Lukomskii, M. D. Mushko

Saratov State University, 83, Astrakhanskaya Str., Saratov, 410012, Russia

Abstract: The classical B-spline is defined recursively as the convolution $B_{n+1}=B_n*B_0$, where $B_0$ is the characteristic function of the unit interval. The classical B-spline is a refinable function and satisfies the Riesz inequality. Therefore any B-spline $B_n$ generates the Riesz multiresolution analysis (MRA). We define binary B-splines, obtained by double integration of the third Walsh function. We give an algorithm for constructing an interpolating spline of the second degree for a binary node system and find the approximation order of this interpolation process. We also prove that the system of dilations and shifts of the constructed B-spline generates an MRA $(V_n)$ in De Boor sense. This MRA is not Riesz. But we can find the approximation order of functions from the Sobolev spaces $W_2^s, s>0$ by the subspaces $(V_n)$.

Key words: binary B-splines, multiresolution analysis, Sobolev spaces.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00152_à This work was supported by the Russian Foundation for Basic Research (project no. 16-01-00152).

DOI: https://doi.org/10.18500/1816-9791-2018-18-2-172-182

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Bibliographic databases:

UDC: 517.51

Citation: S. F. Lukomskii, M. D. Mushko, “On binary B-splines of second order”, Izv. Saratov Univ. Math. Mech. Inform., 18:2 (2018), 172–182

Citation in format AMSBIB
\Bibitem{LukMus18} \by S.~F.~Lukomskii, M.~D.~Mushko \paper On binary B-splines of second order \jour Izv. Saratov Univ. Math. Mech. Inform. \yr 2018 \vol 18 \issue 2 \pages 172--182 \mathnet{http://mi.mathnet.ru/isu753} \crossref{https://doi.org/10.18500/1816-9791-2018-18-2-172-182} \elib{https://elibrary.ru/item.asp?id=35085047}