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.


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


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