On the Ultimate Peano Derivative

✍ Scribed by R.E Svetic; H Volkmer

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 190 KB
• Volume: 218
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1997.5805

✦ Synopsis

A function f : ‫ޒ‬ ª ‫ޒ‬ is said to have an nth Generalized Peano Derivative Ž . GPD at x if f is continuous in a neighborhood of x and there exists an integer Ž . kG0 such that the kth primitive of f has a k q n th Peano derivative at x. An example shows that sometimes no such k exists. In this case, C.-M. Lee has proposed a further generalization when the sequence of derivates, indexed by k, converges to a common value. This value is termed the nth Ultimate Peano Ž . Derivative UPD at x. Here we show that these generalizations of the Peano derivative are related to a certain Laplace integral for which the Tauberian theorem shows that any finite UPD is in fact a GPD.