Direct and inverse estimates for Phillips type operators

In the present paper we define the generalization of the Phillips operators. Direct and converse theorems are established for Phillips operators and its new generalization. The direct theorems are given for the second order Ditzian–Totik modulus of smoothness. The inverse results are derived in the terminology of [J. Anal. Math. 61 (1993) 61–111].  2004 Elsevier Inc. All rights reserved.