Strong inequalities for Hermite-Fejér interpolations and characterization of \(K\)-functionals (Q1724941)

Summary: The works of \textit{S. Smale} and \textit{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)], \textit{F. Cucker} and \textit{S. Smale} [Found. Comput. Math. 2, No. 4, 413--428 (2002; Zbl 1057.68085)], and \textit{F. Cucker} and \textit{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.