# IJPAM: Volume 71, No. 1 (2011)

ON THE CONVOLUTION EQUATION
RELATED TO THE KLEIN-GORDON OPERATOR

Amphon Liangprom, Kamsing Nonlaopon
Department of Mathematics
Khon Kaen University
Khon Kaen, 40002, THAILAND

Abstract. In this paper, we study the distribution , where is the Klein-Gordon operator iterated times defined by (), is a non-negative integer, is the Dirac-delta distribution, is a non-negative real number, is a variable and is a constant and both are the points in the -dimensional Euclidean spaces .

At first, the properties of are studied and after that we study the application of for solving the solution of the convolution equation

where is the generalized function and is a constant. It found that the type of solutions of this convolution equation, such as the ordinary function and the singular distribution depend on the relationship between the values of and .

AMS Subject Classification: 46F10, 46F12

Key Words and Phrases: convolution equation, tempered distribution, Klein-Gordon operator, Dirac-delta distribution