The average errors for Hermite-Fejér interpolation on the Wiener space

Abstract

For 1 ⩾ p < ∞, firstly we prove that for an arbitrary set of distinct nodes in [−1, 1], it is impossible that the errors of the Hermite-Fejer interpolation approximation in L p -norm are weakly equivalent to the corresponding errors of the best polynomial approximation for all continuous functions on [−1, 1]. Secondly, on the ground of probability theory, we discuss the p-average errors of Hermite-Fejer interpolation sequence based on the extended Chebyshev nodes of the second kind on the Wiener space. By our results we know that for 1 ⩽ p < ∞ and 2 ⩽ q < ∞, the p-average errors of Hermite-Fejer interpolation approximation sequence based on the extended Chebyshev nodes of the second kind are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence for L q -norm approximation. In comparison with these results, we discuss the p-average errors of Hermite-Fejer interpolation approximation sequence based on the Chebyshev nodes of the second kind and the p-average errors of the well-known Bernstein polynomial approximation sequence on the Wiener space.