An equivalence theorem on best approximation of continuous functions by algebraic polynomials

Abstract

In the theory of best approximation of continuous functions f(x) on [-1,1] by algebraic polynomials it is well known that ω₂(f^(r),t)=O(t ^α),0<α<2, if and only if there exists a sequence {p _n (x)}₁ ^∞ of algebraic polynomials p _n (x) of order ≦n such that , where . Furthermore, concerning simultaneous approximation of f and its derivatives it is known that the letter assertion implies for every k>r+α. In this paper it is shown that in case of the particular sequence defined by , these three assertions are equilavent to another and moreover to a fourth assertion of Zamansky's type, provided f belongs to some Lipschitz space Lip(β,l) for β>(r+α)/2, i.e. ω_i(f,t)=O(t ^β). This equivalence theorem represents an algebraic counterpart of a theorem on trignometric best approximation in Butzer-Scherer (Aequationes Mathematicae, 3(1969) pp. 170-185).