Convergence of Hermite-Fejér Interpolation via Korovkin’s Theorem

Abstract

From the concept of Hermite-Fejér interpolation there result interpolatory proofs of the Weierstraß approximation theorem for continuous functions of one or several real variables. The one-dimensional problem was posed and solved by L. Fejér [1] in 1916 in case of Chebyshev nodes. Later on in 1930 G. Szegö [11, 12] treated the case of arbitrary Jacobian nodes. About 1960 new ideas came in by the Korovkin theorems, but only for Jacobian nodes i.e. for zeros of the Jacobi polynomials P_m ^(α,β) with max (α,β) ≤ 0 where the Hermite-Fejér. operator is a positive one. Shisha-Mond [9] considered in 1965 the problem of multidimensional Hermite-Fejér interpolation using Chebyshev nodes and tensor product methods. The multidimensional case was also treated in detail by Haußmann [3] and Haußmann-Pottinger [4]. Knoop [5, 6] introduced a smoothing concept to enlarge the possible parameter are (α,β) of the Jacobi polynomials P_m ^(α,β) for which one gets uniform convergence.