On multivariate Baskakov operator

In this paper, we study multivariate Baskakov operator Bn,d (f, x). We first show that the operator can retain some properties of the original function f , such as monotony, semi-additivity and Lipschitz condition, etc. Secondly, we discuss the monotony on the sequence of multivariate Baskakov operator Bn,d (f, x) for n when the function f is convex. Then, we propose, for estimating the rate of approximation, a new modulus of smoothness and prove the modulus to be equivalent to certain K-functional. Finally, with the modulus of smoothness as metric, we establish a strong direct theorem by using a decomposition technique for the operator.  2004 Elsevier Inc. All rights reserved