Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 709607, 16 pages
Research Article

On the Polyconvolution with the Weight Function for the Fourier Cosine, Fourier Sine, and the Kontorovich-Lebedev Integral Transforms

Faculty of Applied Mathematics and Informatics, Hanoi University of Technology, No. 1, Dai Co Viet, Hanoi, Vietnam

Received 26 December 2009; Revised 5 April 2010; Accepted 17 May 2010

Academic Editor: Slimane Adjerid

Copyright © 2010 Nguyen Xuan Thao. 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.


The polyconvolution with the weight function γ of three functions f,g, and h for the integral transforms Fourier sine (Fs), Fourier cosine (Fc), and Kontorovich-Lebedev (Kiy), which is denoted by γ(f,g,h)(x), has been constructed. This polyconvolution satisfies the following factorization property Fc(γ(f,g,h))(y)=sin y(Fsf)(y)(Fcg)(y)(Kiyh)(y), for all y>0. The relation of this polyconvolution to the Fourier convolution and the Fourier cosine convolution has been obtained. Also, the relations between the polyconvolution product and others convolution product have been established. In application, we consider a class of integral equations with Toeplitz plus Hankel kernel whose solution in closed form can be obtained with the help of the new polyconvolution. An application on solving systems of integral equations is also obtained.