International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 97250, 11 pages
doi:10.1155/2007/97250
Research Article

Integral Transforms of Fourier Cosine and Sine Generalized Convolution Type

Nguyen Xuan Thao,1 Vu Kim Tuan,2 and Nguyen Thanh Hong3

1Hanoi Water Resources University, 175 Tay Son, Dong Da, Hanoi, Vietnam
2Department of Mathematics, University of West Georgia, Carrollton, GA30118, USA
3Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam

Received 5 April 2007; Accepted 27 August 2007

Academic Editor: Piotr Mikusinski

Copyright © 2007 Nguyen Xuan Thao et al. 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

Integral transforms of the form f(x)↦g(x)=(1−d2/dx2){∫0∞k1(y)[f(|x+y−1|)+f(|x−y+1|)−f(x+y+1)−f(|x−y−1|)]dy+∫0∞k2(y)[f(x+y)+f(|x−y|)]dy} from Lp(ℝ+) to Lq(ℝ+), (1≤p≤2,p−1+q−1=1) are studied. Watson's and Plancherel's theorems are obtained.