×

Strong inequalities for Hermite-Fejér interpolations and characterization of \(K\)-functionals. (English) Zbl 1475.41004

Summary: The works of S. Smale and D.-X. Zhou [Anal. Appl., Singap. 1, No. 1, 17–41 (2003; Zbl 1079.68089); Constr. Approx. 26, No. 2, 153–172 (2007; Zbl 1127.68088)], F. Cucker and S. Smale [Found. Comput. Math. 2, No. 4, 413–428 (2002; Zbl 1057.68085)], and F. Cucker and D. X. Zhou [Learning theory. An approximation theory viewpoint. Cambridge: Cambridge University Press (2007; Zbl 1274.41001)] indicate that approximation operators serve as cores of many machine learning algorithms. In this paper we study the Hermite-Fejér interpolation operator which has this potential of applications. The interpolation is defined by zeros of the Jacobi polynomials with parameters \(- 1 < \alpha\), \(\beta < 0\). Approximation rate is obtained for continuous functions. Asymptotic expression of the \(K\)-functional associated with the interpolation operators is given.

MSC:

41A05 Interpolation in approximation theory
41A40 Saturation in approximation theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Zhou, D.-X.; Jetter, K., Approximation with polynomial kernels and SVM classifiers, Advances in Computational Mathematics, 25, 1-3, 323-344 (2006) · Zbl 1095.68103 · doi:10.1007/s10444-004-7206-2
[2] Szabados, J.; Vértesi, P., Interpolation of Functions (1990), World Scientific · Zbl 0721.41003 · doi:10.1142/9789814335843
[3] Szabados, J., On the convergence and saturation problem of the Jackson polynomials, Acta Mathematica Academiae Scientiarum Hungaricae, 24, 399-406 (1973) · Zbl 0269.42003 · doi:10.1007/BF01958053
[4] Zhou, X. L., On a problem of Szabados, Atti del Seminario Matematico e Fisico dell’Università di Modena, 41, 1, 149-165 (1993) · Zbl 0821.41025
[5] Gonska, H. H.; Knoop, H.-B., On Hermite-Fejér interpolation: a bibliography (1914-1987), Studia Scientiarum Mathematicarum Hungarica, 25, 1-2, 147-198 (1990)
[6] Vértesi, P. O. H., Notes on the Hermite-Fejér interpolation based on the Jacobi abscissas, Acta Mathematica Academiae Scientiarum Hungaricae, 24, 233-239 (1973) · Zbl 0267.41001 · doi:10.1007/BF01894632
[7] Vlasov, V. F., The constructive characteristics of a class of functions, Doklady Akademii Nauk SSSR, 142, 773-775 (1962)
[8] DeVore, R. A.; Lorentz, G. G., Constructive Approximation (1993), Berlin, Germany: Springer, Berlin, Germany · Zbl 0797.41016
[9] Xie, T.; Zhou, X., Hermite-Fejér interpolation operator and characterization of functions, Mathematische Nachrichten, 281, 11, 1651-1663 (2008) · Zbl 1153.41002 · doi:10.1002/mana.200510704
[10] Szegő, G., Orthogonal Polynomials, 23 (1975), Providence, RI, USA: American Mathematical Society, Providence, RI, USA
[11] Zhou, X. L., Hermite-Fejér-type operators. I, Approximation Theory and Its Applications, 10, 4, 37-55 (1994) · Zbl 0867.41001
[12] Zhou, X. L.; Tandori, K.; Szabados, J., Weighted approximation by Hermite-Fejér interpolation, Approximation Theory (Proceedings International Conference in Kecskemet, Hungary, 1990). Approximation Theory (Proceedings International Conference in Kecskemet, Hungary, 1990), Colloq. Mathematical Society János Bolyai, 58, 785-798 (1991), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0777.41005
[13] Xie, T.; Zhou, X., Lower bounds of some singular integrals and their applications, Archiv der Mathematik, 88, 3, 249-258 (2007) · Zbl 1117.42002 · doi:10.1007/s00013-006-1864-x
[14] Lorentz, G. G., Approximation of Functions (1966), New York, NY, USA: Holt, Rinehart and Winston, New York, NY, USA · Zbl 0153.38901
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.