# IJPAM: Volume 52, No. 2 (2009)

THE ALGEBRA OF SMOOTH FUNCTIONS
OF RAPID DESCENT

Paul O. Oleche, N. Omolo-Ongati, John O. Agure
Department of Mathematics
Maseno University
P.O. Box 333, Maseno, KENYA
e-mail: poleche@maseno.ac.ke
e-mail: omolo_ongati@yahoo.com
e-mail: johnagure@maseno.ac.ke

Abstract.A bounded operator with the spectrum lying in a compact set , has  functional calculus. On the other hand, an operator acting on a Hilbert space H, admits a  functional calculus if is self-adjoint. So in a Banach space setting, we really desire a large enough intermediate topological algebra A, with such that spectral operators or some sort of their restrictions, admit a A functional calculus.

In this paper we construct such an algebra of smooth functions on R that decay like as , for some . Among other things, we prove that is dense in this algebra. We demonstrate that important functions like are either in the algebra or can be extended to functions in the algebra. We characterize this algebra fully.