A representation theorem and approximation operators arising from inequalities involving differential operators

Abstract

Abstract. A representation of functions as integrals of a kernel <ji(t; x), which was introduced by Studden, with respect to functions of bounded variation in [0, oo) is obtained whenever the functions satisfy some conditions involving the differential operators (<//<# ){/(z)/>v¡(r)}, z = 0, 1,2,....The results are related to the concepts of generalized completely monotonie functions and generalized absolutely monotonie functions in (0, co). Some approximation operators for the approximation of continuous functions in [0, oo) arise naturally and are introduced; some sequenceto-function summability methods are also introduced.