ON THE ULTIMATE PEANO DERIVATIVE

A function f: R --> R is said to have an nth Generalized Peano Derivat ive (GPD) at x if f is continuous in a neighborhood of x and there exi sts an integer k greater than or equal to 0 such that the kth primitiv e of f has a (k + n)th Peano derivative at x. An example shows that so metimes no such k exists. In this case, C.-M. Lee has proposed a furth er generalization when the sequence of derivates, indexed by k, conver ges to a common value. This value is termed the nth Ultimate Peano Der ivative (UPD) at x. Here we show that these generalizations of the Pea no derivative are related to a certain Laplace integral for which the Tauberian theorem shows that any finite UPD is in fact a GPD. (C) 1998 Academic Press.