Peano derivatives and general integrals

Peter Southcott Bullen and S. N. Mukhopadhyay

Vol. 47 (1973), No. 1, 43–58

DOI: 10.2140/pjm.1973.47.43

Abstract

It is known that if a function f has a finite derivative (or approximate derivative) on a set E on which f is continuous then f is ACG_∗ (or ACG) on E and that if f is ACG_∗ (or ACG) on a set E then a finite derivative (or approximate derivative) of f exists almost everywhere in E. These results are extended by Sargent in the case of generalized derivatives of higher order. She has proved that if f_n+1, the generalized derivative of f of order n + 1, exists in an interval [a,b] then the derivative f_n is V _n − ACG^∗ on [a,b] and that if f_n is V _n − ACG^∗ on [a,b] then f_n+1 exists and is equal to the approximate derivative of f_n almost everywhere in [a,b].

The present work is concerned with extending still further these results of Sargent by introducing a more general definitions of absolute continuity for the n-th derivatives. It also introduces an approximate Pⁿ-integral which generalizes the Pⁿ-integral of James and Bullen.

Mathematical Subject Classification 2000

Primary: 26A24

Milestones

Received: 1 May 1972

Published: 1 July 1973