Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Chumachenko S. A. Smooth Approximations in C[0, 1]. Izv. Sarat. Univ. Math. Mech. Inform., 2020, vol. 20, iss. 3, pp. 326-342. DOI: 10.18500/1816-9791-2020-20-3-326-342

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2020
Full text:
(downloads: 268)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
501.1
DOI: 
10.18500/1816-9791-2020-20-3-326-342

Smooth Approximations in C[0, 1]

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

The first orthonormal basis in the space of continuous functions was constructed by Haar in 1909. In 1910, Faber integrated the Haar system and obtained the first basis of continuous functions in the space of continuous functions. Schauder rediscovered this system in 1927. All functions of Faber – Shauder are piecewise linear, and partial sums are inscribed polygons. There was many attempts to build smooth analogues of the Faber – Schauder basis. In 1965, K. M. Shaidukov succeeded. The functions he constructed were smooth, but consisted of parabolic arcs. Shaidukov proved the uniform convergence of the obtained expansions, but failed to obtain deviation estimates. Another analogue of the Faber – Schauder system was proposed by T. U. Aubakirov and N. A. Bokaev in 2007. They built a class of functions that form a basis in the space of continuous functions, obtained estimates of the deviation of partial sums from the approximated function. The functions constructed were piecewise linear, as in the Faber – Schauder system. We construct smooth analogues of the Faber – Schauder system and obtain estimates of the deviation of partial sums from the approximate function. Those are systems of compressions and shifts of a single function, which we call the binary basic spline. The binary basic spline is an integral of the n-th order of the Walsh function W2n −1. Thus, we were able to construct analogues of the Faber – Schauder system with a large degree of smoothness and obtain deviations in terms of module of continuity.

References: 
  1. Faber G. Uber die ortogonalenfunctionen des Herrn Haar. Jahresber. Deutsch Math. Verein., 1910, vol. 19, pp. 104–112.
  2. Schauder J. Zur Theorie stetiger Abbildungen in Fimktionalraumen. Math. Z., 1927, Bd. 26, pp. 47–65.
  3. Matveev V. A. On Schauder system series. Math. Notes, 1967, vol. 2, iss. 3, pp. 646–652. DOI: https://doi.org/10.1007/BF01094054
  4. Ciesielski Z. Some properties of Schauder basis of space C[0, 1]. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 1960, vol. 3, pp. 141–144.
  5. Bochkarev S. V. Series with respect to the Schauder system. Math. Notes, 1968, vol. 4, iss. 4, pp. 763–767. DOI: https://doi.org/10.1007/BF01093716
  6. Ulianov P. L. On series with respect to a Schauder system. Math. Notes, 1970, vol. 7, iss. 4, pp. 261–268. DOI: https://doi.org/10.1007/BF01151699
  7. Saburova T. N. Composite functions and their series in the Faber – Schauder system. Math. USSR-Izv., 1972, vol. 6, iss. 2, pp. 395–415. DOI: https://doi.org/10.1070/IM1972v006n02ABEH001879
  8. Goryachev A. P. The Fourier coefficients with respect to the Faber – Schauder system. Math. Notes, 1974, vol. 15, iss. 2, pp. 192–198. DOI: https://doi.org/10.1007/BF02102406
  9. Abramova V. V. Faber – Schauder system on the triangular mesh. Mathematics. Mechanics, 2015, iss. 17, pp. 3–6 (in Russian).
  10. Kashin B. S., Saakian A. A. Ortogonal’nye riady [Ortogonal series]. Moscow, AFC, 1999. 550 p. (in Russian).
  11. Aubakirov T. U., Bokaev N. A. On a new class of function systems of Faber – Schauder type. Math. Notes, 1974, vol. 82, iss. 5, pp. 643–651. DOI: https://doi.org/10.1134/S0001434607110016
  12. Shaidukov K. M. Parabolic basis in continuos function space. Uchenie zapiski Kazanskogo universiteta, 1965, vol. 125, no. 2, pp. 133–142 (in Russian). DOI: https://doi.org/10.4213/mzm3840
  13. Lukomskii S. F., Terekhin P. A., Chumachenko S. A. Rademacher Chaoses in Problems of Constructing Spline Affine Systems. Math. Notes, 2018, vol. 103, iss. 6, pp. 919–928. DOI: https://doi.org/10.1134/S0001434618050280
  14. Chumachenko S. A. One analogue of Faber – Schauder system. Trudy matematicheskogo tsentra imeni Lobachevskogo. Kazan, 2016, vol. 53, pp. 163–164 (in Russian).
Received: 
18.12.2019
Accepted: 
01.04.2020
Published: 
31.08.2020