Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Chumachenko S. A. Binary basic splines in MRA. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 4, pp. 458-471. DOI: 10.18500/1816-9791-2021-21-4-458-471, EDN: XBGXJS

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.11.2021
Full text:
(downloads: 814)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
517.98
EDN: 
XBGXJS

Binary basic splines in MRA

Autors: 
Chumachenko Sergei A., Saratov State University
Abstract: 

$B$-splines were introduced by Carry and Schoenberg. Constructed on a uniform mesh and defined in terms of convolutions, such splines generate a Riesz MRA. We constructed splines $varphi_n$, where $n$ is the order of integration of the Walsh function with the number $2^n - 1$. We called these splines binary basic splines. We know that binary basic splines form a basis in the space of functions that are continuous on the segment $[0, 1]$ and $0$ outside of it. We proved that binary basic splines are a scaling function and generate an MRA of $(V_n)$ which is not a Riesz MRA. The order of approximation was determined by subspaces from Sobolev spaces.

References: 
  1. Schoenberg I. J. On spline functions (with a supplement by T. N. E. Greville). In: O. Shisha, ed. Inequalities I. New York, Academic Press, 1967, pp. 255–291.
  2. de Boor C. A Practical Guide to Spline. (American Mathematical Society, vol. 27). New York, Springer-Verlag, 1978. 348 p. (Russ. ed.: Moscow, Radio i svyaz’, 1985. 304 p.).
  3. Ahlberg J. H., Nilson E. N., Walsh J. L. The Theory of Splines and Their Applications. (Mathematics in Science and Engineering: A Series of Monographs and Textbooks, Vol. 38). Academic Press, 1967. 296 p. (Russ. ed.: Moscow, Mir, 1972. 320 p.).
  4. Kashin B. S., Saakian A. A. Ortogonal’nye riady [Ortogonal Series]. Moscow, AFC, 1999. 550 p. (in Russian).
  5. Novikov I. Ya., Protasov V. Yu., Skopina M. A. Wavelet Theory. (Translations of Mathematical Monographs, vol. 239). Providence, American Mathematical Society, 2011. 506 p. (Russ. ed.: Moscow, Fizmatlit, 2006. 616 p.).
  6. Battle G. A block spin construction of ondelettes. Part 1: Lemarie functions. Communications in Mathematical Physics, 1987, vol. 110, iss. 4, pp. 601–615. https://doi.org/10. 1007/BF01205550
  7. Lemarie P.-G., Meyer Y. Ondelettes et bases Hilbertiennes. Revista Matematica Iberoamericana, 1986, vol. 2, iss. 1–2, pp. 1–18.
  8. Chumachenko S. A. One analogue of Faber – Schauder system. Trudy matematicheskogo tsentra imeni N. I. Lobachevskogo. Vol. 53. Kazan, 2016, pp. 163–164 (in Russian).
  9. Chumachenko S. A. Binary-scaling spline functions. Trudy matematicheskogo tsentra imeni N. I. Lobachevskogo. Vol. 54. Kazan, 2017, pp. 403 (in Russian).
  10. Lukomskii S. F., Mushko M. D. On binary B-splines of second order. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 2, pp. 172–182 (in Russian). https://doi.org/10.18500/1816-9791-2018-18-2-172-182
  11. Lukomskii S. F., Terekhin P. A., Chumachenko S. A. Rademacher chaoses in problems of constructing spline affine systems. Mathematical Notes, 2018, vol. 103, iss. 6, pp. 863–874. https://doi.org/10.4213/mzm11654
  12. Chumachenko S. A. Smooth approximation in C[0, 1]. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 3, pp. 326–342 (in Russian). https://doi.org/10.18500/1816-9791-2020-20-3-326-342
  13. Zhao H. Mathematics in Image Processing. IAS/Park City Mathematics Series, 2013, vol. 19. 245 p. https://doi.org/10.1090/pcms/019
Received: 
13.06.2021
Accepted: 
24.07.2021
Published: 
30.11.2021