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The generalized convolutions related to the fourier sine and Kontorovich - Lebedev integral transformations

In this paper, we establish several properties in a two parametric family of Lebesgue spaces for certain convolution related to the Fourier sine and Kontorovich - Lebedev integral transformations. Norm estimations in the weighted Lp - spaces and applications to solve a class of the integral equations and systems of integral equations are obtained. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2011 Vol. 56 No. 7 pp. 3-13 THE GENERALIZED CONVOLUTIONS RELATED TO THE FOURIER SINE AND KONTOROVICH - LEBEDEV INTEGRAL TRANSFORMATIONS Nguyen Xuan Thao and Nguyen Tien Trung Hanoi University of Science and Technology E-mail tientrungbk@gmail.com Abstract. In this paper we establish several properties in a two parametric family of Lebesgue spaces for certain convolution related to the Fourier sine and Kontorovich - Lebedev integral transformations. Norm estimations in the weighted Lp - spaces and applications to solve a class of the integral equations and systems of integral equations are obtained. Keywords Convolution weight function Kontorovich - Lebedev Fourier transform. 1. Introduction The convolutions for the integral transform have been studied since the be- ginning of the 20th century such as convolutions for the Fourier Laplace Mellin Hankel etc. Theory of convolutions have many interesting applications in solving problems of mathematical physics inverse problems integral equations differential equations and it attracted the attention of many mathematicians. In 1998 Kachichev V.A. and Thao N.X. proposed a method for constructing convolutions with a weight function for three integral transformations. In their recent article in a prestigious journal 5 Yakubovich S.B. and Britvinar L.E. presented their research on gener- alized convolutions whose factorization identity contained two integral transforma- tions Kontorovich - Lebedev and Fourier cosine. In this paper we construct and study two generalized convolutions whose factorization equation contains two in- tegral transformations Kontorovich - Lebedev and Fourier sine. Our study results together with those demonstrated in 5 will give an overview of the classes of con- volution with Fourier and Kontorovich - Lebedev transformations. 3 Nguyen Xuan Thao and Nguyen Tien Trung 2. Well-known result 2.1. Fourier sine tranformation The Fourier sine integral transformation .