Some Remarks on Equivalence of Moduli of Smoothness

Abstract

The present paper investigates polynomials for which the inverse inequality for moduli of smoothness holds. The technique for approach is different from the previous works for splines and is elegantly organized. © 2001 Elsevier Science Key Words: equivalence; modulus of smoothness; polynomial. a, b] ), and w k (f, t) [a, b] be the modulus of smoothness of order k of f ¥ C [a, b] , as usual. We will write w(f, t)=w(f, t) [a, b] for convenience if there is no confusion. Let f(x) be a continuous function on the interval [a, b] which hasIt is well known that a, b] , where w 0 (f, t)=||f|| [a, b] The inverse result of the above inequality does not hold in general. However, for some functions f ¥ C [a, b] , one has t k w m − k (f (k) , t) [ Cw m (f, t) (1)