A Generalized Integral II

Abstract

The definition and some of the properties of what may be called a Perron second integral (P2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D2F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.

A survey of intergration by parts for Perron integrals

“…; in addition </ > is continuous, smooth, ACG, and <j>' exists a.e. and is D-integrable; see James [18], Sklyarenko [35]. Further the above discussion suggests a close connection between the P 2 -and SCP-integrals; this is given by the following theorem due to Cross [8], and Sklyarenko [36].…”

“…If, however, G'(x) = oo, then the second term on the right of (18) shows that lDi? (x) = -oo is possible.…”

“…Although a second order integral is then obtained the development follows that of Definition 1 except that now the requirement that M-m be monotonic, Paragraph 8 (ii), is replaced by M -m being convex. In this way we obtain the James P 2 -integral; see Gage and James [12], James [18]. As in Paragraph 13 we only define the major functions.…”

On the regularity of thePn-integral and its application to summable trigonometric series

“…For, because of the presence of the exceptional set E in condition (v) and (vi) of the definition of major function we cannot apply directly Theorem 3.2 to prove that Q -q is a 2m-convex function for arbitrary major and minor functions Q and q respectively. (As the definition of the P 2m -integral in [9] and that of the P 2 -integral in [7] are also affected by the exceptional sets S and E Q respectively, (see [9] and [7]) they would also need this clarification; but the definition of the P 2 -integral in [6] is not affected since the smoothness of major and minor functions is assumed everywhere). We shall follow the method adopted in [15].…”

“…So, we assume that the theorem is true for m = m 0 i.e., Theorem 3.1, 2m 0 is true and we prove that Theorem 3.1, 2 (m 0 + 1) is also true and so the theorem will be proved to be true for all m by induction on m. We require the following auxiliary lemmas: 6], where…”