On Powers of the Compositions Involving Dirac-Delta and Infinitely Differentiable Functions

ÖZÇAĞ E.

RESULTS IN MATHEMATICS, vol.73, no.1, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 73 Issue: 1
  • Publication Date: 2018
  • Doi Number: 10.1007/s00025-018-0766-0
  • Journal Name: RESULTS IN MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Hacettepe University Affiliated: Yes

Abstract

The symbol delta(k) (f(x)) for an infinitely differentiable function f having a simple root is meaningless in the theory of Schwartz distributions. In this work we first of all give meaning to the symbol delta(k) (f(x)) via neutrix calculus due to van der Corput (J Anal Math 7: 291-398, 1959). Then we consider the case that f is the s-th power of a function g(x) having a simple single root and the particular case delta(k) (x(+)(lambda)) for lambda > 0, k is an element of N. Finally we also give meaning to the symbols H(delta(k)), H-(r)(delta(k)), delta((r))(delta(k)) and G(delta(k)), where H denotes the Heaviside function and G is a bounded continuous summable function on R.