Interpolating Operators for Multiapproximation

Abstract

Problem statement: There are no simple definitions of operators for best multiapproximation and best one sided multiapproximation which work for any measurable function in L_p for, p>0. This study investigated operators that are good for best multiapproximation and best one sided multiapproximation. Approach: We first introduced some direct results related to the approximation problem of continuous functions by Hermit-Fejer interpolation based on the zeros of Chebyshev polynomials of the first or second kind in terms of the usual modulus of continuity. They were then improved to spaces L_p for p<1, in terms of the first order averaged modulus of smoothness. However, because this suffers from convergence problems, we improved and generalized these direct estimations by defining an interpolating multivariate operator H_n(f) of measurable functions, that operator based on the zeros of Chepyshev polynomials of the first kind and prove that for any measurable function defined on L_p[-1,1 ]^d the sequence H_n(f) converges uniformly to f. Results: The resulting operators were defined for functions f such that f^(k), k = 0,1,… is of bounded variation. Then, the order of best onesided trigonometric approximation to bounded measurable functions in terms of the average modulus of smoothness was characterized. Estimates characterizing the order of best onesided approximation in terms of the k-th averaged modulus of smoothness for any function in spaces L_p, p< ∞ were obtained. In our research we also approximated one sidedly these measurable functions in L_p[-1,1]^d by defining a new operator for onesided approximation and prove a direct theorem for best one sided multiapproximation in terms of the first order averaged moduli of smoothness. Conclusion: The proposed method successfully construct operators for best multi approximation and best one sided multiapproximation for any measurable function in L_p for, p>0.