STRONG CONVERSE INEQUALITIES

Converse inequalities are proved for a family of operators that state the equivalence of two terms of error in approximation to the revelant modulus of smoothness. Such inequalities have been proved by Z. Ditzi an and K. G. Ivanov with a different method. Our emphasis is that thes e so-called strong converse inequalities follow from some standard est imates on the derivatives of the operators without additional work; he nce we extend the Ditzian-Ivanov result to a large family of operators . The method of the paper is very close in spirit to the classical par abola technique. (C) 1994 Academic Press, Inc.