A New Modulus of Smoothness for Uniform Approximation

Abstract

The estimates of best approximation using classical modulus of smoothness is not uniform. Also we sometimes need to improve the degree of best approximation near the end points. Thus we need to improve this classical modulus of smoothness. Here we define a new modulus of smoothness to achieve uniform estimates of  best approximation and an improvement of  a degree of such version of best approximation.  Our modulus of smoothness is for k-monotone functions. Estimates for using our  modulus of smoothness are introduced. Applications for these estimates are also introduced