Cite this article as:

Lukomskii S. F., Mushko M. D. On Binary B-splines of Second Order. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 2, pp. 172-182. DOI: https://doi.org/10.18500/1816-9791-2018-18-2-172-182

Language: 
Russian
Heading: 
Mathematics
UDC: 
517.51

On Binary B-splines of Second Order

Authors: 
Lukomskii Sergei Feodorovich, Saratov State University 1, Saratov State University, Russia, 410012, Saratov, Astrahanskaya st., 83 , LukomskiiSF@info.sgu.ru
Mushko Maxim D. , Saratov State University, Russia, Saratov, Astrakhanskaya 83 , dartmaximus@yandex.ru
Abstract: 
The classical B-spline is defined recursively as the convolution B n+1= Bn∗ B0, where B0 is the characteristic function of the unit interval. The classical B-spline is a refinable function and satisfies the Riesz inequality. Therefore any B-sp lin e Bn 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 in terpolation process. We also prove that the system of dilations and shifts of the constructed B-spline generates an MRA (Vn) in De Boor sense. This MRA is not Riesz. But we can find the approximation order of function s from the Sobolev spaces Ws2, s > 0 by the subspaces (Vn).
Key words: 
bina ry B-splines, multiresolution analysis, Sobolev spaces
DOI: 
https://doi.org/10.18500/1816-9791-2018-18-2-172-182
References

1. Curry H. B., Schoenberg I. J., On spline distributions and their limits: the Pollya distributions. Bull. Amer. Math. Soc., 1947, vol. 53, Abstract 380t, p. 1114.
2. Schoenberg I. J. On spline functions (with a supplement by T. N. E. Greville). Inequalities I. Ed. O. Shisha. New York, Academic Press, 1967, pp. 255–291.
3. Schoenberg I. J. Contributions to problem of approximation of equidistant data by analytic functions. Quart. Appl. Math., 1946, vol. 4, pp. 45–99, 112–141.
4. Alberg J. H., Nilson E. N., Walsh J. L. T he theory of splines and their Applications. Academic Press, 1967. 296 p.
5. De Boor C. A practical guide to splines. New York, Springer-Verlag, 2001. 348 p. (Russ.ed.: Moscow, Radio i sviaz’, 1985. 304 p.)
6. Str¨omberg J.-O. A modified Franklin system and higher-order spline systems on Rn as unconditional bases for Hardy spaces. Conference in Harmonic Analysis in Honor of A.Zigmund (The Wadsworth Mathematics Series). Eds. W. Beckner, A. P. Calderon. Springer, 1982, vol. 2, pp. 475–494.
7. Battle G. A block spin construction of ondelettes. Part 1: Lemarie functions. Comm. Math. Phys., 1987, vol. 110, pp. 601–615.
8. Lemarie P.-G., Meyer Y. Ondelettes et bases Hilbertiennes. Rev. Math. Iber., 1987, vol. 2, no. 1/2, pp. 1–18.
9. Chumachenko S. On an analogue of the Faber – Schauder system. Trudy matematicheskogo centra N. I. Lobachevsky [Proceedings of the N. I. Lobachevsky Mathematical Center]. 2016, vol. 53, pp. 163–164 (in Russian).
10. Mathematics in image processing. Ed. Hongkai Zhao. IAS/Park City Mathematics Series. 2013, vol . 19. 245 p.
11. De Boor C., DeVore R. A., Ron A. Approximation from shift-invariant subspaces of L2(Rd). Transactions of the American Mathematical Society, 1994, vol. 341, no. 2, pp. 787–806.
12. De Boor C., DeVore R. A., Ron A. On the construction of multivariante (pre) wavelets. Constructive approximation, 1993, vol. 9, no. 2, pp. 123–166.
13. Jia R. Q., Shen Z. Multiresolution and Wavelets. Proc. Edinb. Math. Soc., II. Ser., 1994, vol. 37, no. 2, pp. 271–300.
14. Jia R. Q., Micchelli C. A. Using the refinement equations for the construction of pre- wavelets II: Powers of two. Curves and surfaces. Eds. P.-J. Laurent, A. Le Mehaute, L. L. Schumaker. Elsevier Inc., 1999, pp. 209–246.
15. Chui Ch. K. An Introduction to Wavelets. San Diego, CA, USA, Academic Press, 1992. 264 p. (Russ. ed.: Moscow, Mir, 2001. 412 p.)

Short text (in English): 
mmi_2018_18_2-eng-7-8.pdf
57
Full text:
download
358