Generalized nth Primitives

Abstract

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The pn-integral

“…This definition differs from that in [5], in the use of the end-points in (a), in the existence of the sets £ and C, and also in condition (d). In [5] the major function is normalized instead by requiring it to be zero on a given set of n distinct points a u ---,a n ; let us call these functions, J-majorfunctions over (a t )igitg n .…”

“…A real-valued function F denned on the closed bounded interval [a,b] is said to be n-convex on [a, b] iff for all choices of n + 1 distinct points, x 0 , ••,x n , in [a, 6], V n (F;x k ) ^ 0, [2,5]. If -F is n-convex then F is said to be n-concave.…”

“…In [5] the major function is normalized instead by requiring it to be zero on a given set of n distinct points a u ---,a n ; let us call these functions, J-majorfunctions over (a t )igitg n .…”

“…His proofs are often indirect, using properties of the QP-integrals of Burkill, [3]. In this paper a simpler definition of the P"-integral is given -the original and not completely equivalent definition, was probably chosen as James considered this integral as a special case of one defined in terms of certain symmetric derivatives, [5], when end points of the interval of definition had naturally to be avoided. We then give direct proofs of the basic results, give a characterization of ^"-primitives, and connect the integral with certain work of Denjoy, [4].…”

A new characterization of Cesàro-Perron integrals using Peano derivates

“…The method is that of Perron using Peano dérivâtes. James [3] has also used Peano dérivâtes to define an integral, the difference here being the use of a premajorant function as well as a majorant. The properties of the integral are all derived from properties of the generalized derivatives of the premajorant and preminorant functions.…”