IJPAM: Volume 80, No. 5 (2012)


A.S. Okb El Bab$^1$, A.M. Zabel$^2$, H.A. Ghany$^{3,4}$
$^{1,2}$ Department of Mathematics
Faculty of Science
Al Azhar University
Naser City, Cairo, EGYPT
$^{3}$Department of Mathematics
Faculty of Science
Taif University
$^4$Department of Mathematics
FIE, Helwan University
Cairo, EGYPT

Abstract. This paper is devoted to give the necessary and sufficient conditions guarantees that the product of two positive definite functions defined in a hypercomplex system $L_1(Q,m)$ is also positive definite in $L_1(Q,m)$. Also, we prove that a continuous function with compact support $\psi$ is negative definite if and only if $\exp(-\alpha\psi)$ is positive definite for each $\alpha>0$. Moreover, we will give integral representations of some harmonic functions related to positive definite functions in hypercomplex system.

Received: August 29, 2012

AMS Subject Classification: 43A62, 43A22, 43A10

Key Words and Phrases: hypercomplex system, positive definite functions

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 80
Issue: 5