On the neutral Hermite product of the distributions x^±n and δ^(m)(x)

Abstract

In this paper, we introduce the neutral Hermite product and study products of tempered distributions based on the Hermite expansions. In particular, we consider products of distributions x ⁿ and δ^(m)(x) by using the neutral Hermite product and the results are identical with the classical theory of Schwartz distributions. Finally, the recurrence relations about the Hermite expansion coefficients of x ^±n δ^(m)(x) are obtained for the numerical computation.

Keywords:

• Hermite function
• Hermite expansions
• product of distributions

Acknowledgements

The authors are grateful to Professor Chin Cheng Chou who helped them in the theory of distributions. The authors are also grateful to Professor J.-A. Marti and A. Delcroix for introducing the algebra of Colombeau generalized functions during the summer of 2004 in Shanghai. This work is supported by National Science Foundation of China.