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

Abstract

A representation of functions as integrals of a kernel ψ ( t ; x ) \psi (t;x) , which was introduced by Studden, with respect to functions of bounded variation in [ 0 , ∞ ) [0,\infty ) is obtained whenever the functions satisfy some conditions involving the differential operators ( d / d t ) { f ( t ) / w i ( t ) } , i = 0 , 1 , 2 , … (d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots . The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in ( 0 , ∞ ) (0,\infty ) . Some approximation operators for the approximation of continuous functions in [ 0 , ∞ ) [0,\infty ) arise naturally and are introduced; some sequence-to-function summability methods are also introduced.