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

**THE ALGEBRA OF SMOOTH FUNCTIONS**

OF RAPID DESCENT

OF RAPID DESCENT

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.

**Received: **February 21, 2009

**AMS Subject Classification: **46J15

**Key Words and Phrases: **Banach algebra, smooth function, extension

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 52

**Issue:** 2