×
Gladkaya, A. V.; Vinogradov, O. L.

Sharp Jackson type inequalities for spline approximation on the axis. (English) Zbl 1399.41012

Anal. Math. 43, No. 1, 27-47 (2017).
Summary: We establish several Jackson type inequalities with explicit constants for spline approximation of functions defined on the real axis. The inequalities for the first modulus of continuity of odd derivatives are sharp. We also obtain inequalities for high-order moduli of continuity of a function itself. One of the inequalities for the second modulus of continuity is sharp. Up to the present paper no estimates for spline approximation on the axis in terms of high-order moduli of continuity, with constants written explicitly, were known.

Cited in 1 Document

MSC:

41A15 Spline approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A44 Best constants in approximation theory

Keywords:

Akhiezer-Krein-Favard-type operator; modulus of continuity; spline; Jackson-type inequality
Cite Review PDF
Full Text: DOI

References:

[1] N. Achyeser and M. Krein, Sur la meilleure approximation des fonctions périodiques dérivables au moyen de sommes trigonométriques, C. R. (Dokl.) Acad. Sci. URSS, 15 (1937), 107–111. · Zbl 0016.30001
[2] N. I. Akhiezer, Lectures on the Theory of Approximation, Nauka (Moscow, 1965) (in Russian).
[3] J. Favard, Sur les meilleurs procédés d’approximation de certaines classes des fonctions par des polynomes trigonométriques, Bull. Sci. Math., 61 (1937), 209–224, 243–256. · JFM 63.0225.01
[4] S. Foucart, Y. Kryakin and A. Shadrin, On the exact constant in the Jackson–Stechkin inequality for the uniform metric, Constr. Approx., 29 (2009), 157–179. · Zbl 1181.41017 · doi:10.1007/s00365-008-9039-6
[5] N. P. Korneichuk, Exact Constants in Approximation Theory, Nauka (Moscow, 1987) (in Russian); translated in Encyclopedia of Mathematics and its Applications, 38, Cambridge University Press (Cambridge,1991).
[6] A. A. Ligun, Exact constants of approximation for differentiable periodic functions, Mat. Zametki, 14:1 (1973), 21–30 (in Russian); translated in Math. Notes, 14:1 (1973), 563–569.
[7] I. J. Schoenberg, Cardinal Spline Interpolation, SIAM (Philadelphia, 1993).
[8] V. V. Shalaev, To the question of approximation of continuous periodic functions by trigonometric polynomials, in: Research on Modern Problems of Summation and Approximation of Functions and Their Applications, vol. 8, Dnepropetrovsk. Gos. Univ. (Dnepropetrovsk, 1977), pp. 39–43 (in Russian).
[9] O. L. Vinogradov, Analog of the Akhiezer–Krein–Favard sums for periodic splines of minimal defect, Problemy Mat. Analiza, 2 (2003), 29–56 (in Russian); translated in J. Math. Sci. (N. Y.), 114 (2003), 1608–1627. · Zbl 1050.41016
[10] O. L. Vinogradov, Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts, Mat. Zametki, 85:4 (2009), 569–584 (in Russian); translated in Math. Notes, 85 (2009), 544–557. · Zbl 1211.41003 · doi:10.4213/mzm4162
[11] O. L. Vinogradov and A. V. Gladkaya, A nonperiodic spline analog of the Akhiezer–Krein–Favard operators, Zapiski nauchnykh seminarov POMI, 440 (2015), 8–35 (in Russian); translated in J. Math. Sci. (N. Y.), 217 (2016), 3–22.
[12] O. L. Vinogradov and V. V. Zhuk, Sharp estimates for deviations of linear approximation methods for periodic functions by linear combinations of moduli of continuity of different order, Problemy Mat. Analiza, 25 (2003), 57–97 (in Russian); ranslated in J. Math. Sci. (N. Y.), 114 (2003), 1628–1656. · Zbl 1050.41018
[13] O. L. Vinogradov and V. V. Zhuk, Estimates for functionals with a known moment sequence in terms of deviations of Steklov type means, Zapiski nauchnykh seminarov POMI, 383 (2010), 5–32 (in Russian); translated in J. Math. Sci. (N. Y.), 178 (2011), 115–131.
[14] O. L. Vinogradov and V. V. Zhuk, Estimates for functionals with a known finite set of moments in terms of moduli of continuity and behavior of constants in the Jackson-type inequalities, Algebra i Analiz, 24:5 (2012), 1–43 (in Russian); translated in St. Petersburg Math. J., 24 (2013), 691–721.
[15] V. V. Zhuk, Approximation of Periodic Functions, Izd. Leningrad. Univ. (Leningrad, 1982) (in Russian). · Zbl 0521.42003
[16] V. V. Zhuk, Some sharp inequalities between best approximations and moduli of continuity, Sibirsk. Mat. Zh., 12:6 (1971), 1283–1291 (in Russian); translated in Sib. Math. J., 12:6 (1971), 924–930.
[17] V. V. Zhuk, On sharp inequalities between best approximations and moduli of continuity, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1 (1974), 21–26 (in Russian).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.