Weighted Moduli of Smoothness and Spline Spaces

AbstractIn this paper we study relations between moduli of smoothness with the step-weight functionϕand the best approximation by splines with knots uniformly distributed according to the measure with density 1/ϕ(x). The direct and converse results are obtained for a class of step-weight functions, containingϕ(x)=x(1−x); it is well known that the modulus of smoothness corresponding to thisϕis related to the best polynomial approximation. As a consequence, we obtain relations between the best polynomial and spline approximations