Optimal subspaces for mean square approximation of classes of differentiable functions on a segment

Authors

  • Oleg L. Vinogradov
  • Anastasiya Yu. Ulitskaya

DOI:

https://doi.org/10.21638/spbu01.2020.304

Abstract

In this paper, we specify a set of optimal subspaces for L2 approximation of three classes of functions in the Sobolev spaces W(r) 2 , defined on a segment and subject to certain boundary conditions. A subspace X of dimension not exceeding n is called optimal for a function class A if the best approximation of A by X equals the Kolmogorov n-width of A. These boundary conditions correspond to subspaces of periodically extended functions with symmetry properties. All of the approximating subspaces are generated by equidistant shifts of a single function. The conditions of optimality are given in terms of Fourier coefficients of a generating function. In particular, we indicate optimal spline spaces of all degrees d > r − 1 with equidistant knots of several different types.

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References

Литература

1. Floater M. S., Sande E. Optimal spline spaces for L2 n-width problems with boundary conditions // Constructive Approximation. 2018. P. 1–18.

2. Kolmogorov A. ¨Uber die beste Ann¨aherung von Funktionen einer gegebenen Funktionenklasse // Ann. Math. 1936. Vol. 37. P. 107–110.

3. Виноградов О.Л., Улицкая А.Ю. Точные оценки среднеквадратичных приближений классов дифференцируемых периодических функций пространствами сдвигов // Вестник С.-Петерб. ун-та. Математика. Механика. Астрономия. 2018. Т. 5 (63). Вып. 1. С. 22–31. https://doi.org/10.21638/11701/spbu01.2018.103

4. Schoenberg I. J. Cardinal Spline Interpolation. 2 ed. Philadelphia: SIAM, 1993.

5. Golomb M. Approximation by periodic spline interpolants on uniform meshes // Journal of Approximation Theory. 1968. Vol. 1. P. 26–65. https://doi.org/10.1016/0021-9045(68)90055-5

6. Kamada M., Toriachi K., Mori R. Periodic spline orthonormal bases // Journal of Approximation Theory. 1988. Vol. 55. P. 27–34. https://doi.org/10.1016/0021-9045(88)90108-6

7. Виноградов О.Л. Аналог сумм Ахиезера-Крейна-Фавара для периодических сплайнов минимального дефекта // Проблемы математического анализа. 2003. Вып. 25. С. 29–56.

8. Виноградов О.Л. Точные неравенства для приближений классов периодических сверток пространствами сдвигов нечетной размерности // Математические заметки. 2009. Т. 85, №4. С. 569–584. https://doi.org/10.4213/mzm4162

References

1. Floater M. S., Sande E., “Optimal spline spaces for L2 n-width problems with boundary conditions”, Constructive Approximation, 1–18 (2018).

2. Kolmogorov A., “¨Uber die beste Ann¨aherung von Funktionen einer gegebenen Funktionenklasse”, Ann. Math. 37, 107–110 (1936).

3. Vinogradov O. L., Ulitskaya A.Yu., “Sharp estimates for mean square approximation of classes of differentiable periodic functions by shift spaces”, Vestnik St. Petersburg University. Mathematics 51, iss. 1, 15–22 (2018). https://doi.org/10.3103/S1063454118010120

4. Schoenberg I. J., Cardinal Spline Interpolation (2 ed., Philadelphia, SIAM, 1993).

5. Golomb M., “Approximation by periodic spline interpolants on uniform meshes”, Journal of Approximation Theory 1, 26–65 (1968). https://doi.org/10.1016/0021-9045(68)90055-5

6. Kamada M., Toriachi K., Mori R., “Periodic spline orthonormal bases”, Journal of Approximation Theory 55, 27–34 (1988). https://doi.org/10.1016/0021-9045(88)90108-6

7. Vinogradov O. L., “Analog of the Akhiezer-Krein-Favard sums for periodic splines of minimal defect”, Journal of Mathematical Sciences 114(5), 1608–1627 (2003). https://doi.org/10.1023/A:1022360711364

8. Vinogradov O. L., “Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts”, Mathematical Notes 85, 544–557 (2009). https://doi.org/10.1134/S0001434609030250

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Published

2020-09-04

How to Cite

Vinogradov, O. L., & Ulitskaya, A. Y. (2020). Optimal subspaces for mean square approximation of classes of differentiable functions on a segment. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(3), 404–417. https://doi.org/10.21638/spbu01.2020.304

Issue

Vol. 7 No. 3 (2020)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.