Abstract and Applied Analysis
Volume 2011 (2011), Article ID 908491, 16 pages
http://dx.doi.org/10.1155/2011/908491
Research Article

On the Convolution Equation Related to the Diamond Klein-Gordon Operator

Amphon Liangprom1 and Kamsing Nonlaopon1,2

1Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 11 July 2011; Revised 3 August 2011; Accepted 31 August 2011

Academic Editor: Elena Braverman

Copyright © 2011 Amphon Liangprom and Kamsing Nonlaopon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the distribution 𝑒 𝛼 𝑥 ( ⋄ + 𝑚 2 ) 𝑘 𝛿 for 𝑚 ≥ 0 , where ( ⋄ + 𝑚 2 ) 𝑘 is the diamond Klein-Gordon operator iterated 𝑘 times, 𝛿 is the Dirac delta distribution, 𝑥 = ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) is a variable in ℝ 𝑛 , and 𝛼 = ( 𝛼 1 , 𝛼 2 , … , 𝛼 𝑛 ) is a constant. In particular, we study the application of 𝑒 𝛼 𝑥 ( ⋄ + 𝑚 2 ) 𝑘 𝛿 for solving the solution of some convolution equation. We find that the types of solution of such convolution equation, such as the ordinary function and the singular distribution, depend on the relationship between 𝑘 and 𝑀 .