On a Polyconvolution with a Weight Function for Fourier Cosine and Laplace Transforms
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On a Polyconvolution with a Weight Function for Fourier Cosine and Laplace Transforms N. M. Khoa1
Received: 17 March 2015 / Revised: 20 April 2015 / Accepted: 25 April 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
Abstract In this paper, we introduce the generalized convolution with a weight function for the Hartley and Fourier cosine transforms. Several algebraic properties and applications of this generalized convolution to solving a class of integral equations of Toeplitz plus Hankel type and a class of systems of integral equations are presented. Keywords Laplace transform · Fourier cosine transform · Convolution · Generalized convolution · Polyconvolution · Integral equations Mathematics Subject Classification (2010) 44A35 · 45E10 · 42A38
1 Introduction Convolutions and generalized convolutions for many different integral transforms have interesting applications in several contexts of science and mathematics ([2, 3, 5, 7–10, 12, 16–18]). In 1997, Kakichev ([4]) proposed a general definition of polyconvolution for n + 1 arbitrary integral transforms T , T1 , T2 , . . . , Tn with the weight function γ (x) of functions f1 , f2 , . . . , fn for which the factorization property holds γ T ∗ (f1 , f2 , . . . , fn ) (y) = γ (y) (T1 f1 ) (y) (T2 f2 ) (y) . . . (Tn fn ) (y) . An application of this notion to three integral transforms as Fourier, Fourier cosine, Fourier sine, or Hartley and types of Fourier transforms has been presented ([6, 11]). The generalized convolution generated by the Fourier cosine transform and the Laplace transform has been studied in [13–15]. Following these authors, in this paper, we construct and
N. M. Khoa
[email protected] 1
Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
N. M. Khoa
study a new polyconvolution with a weight function for a bunch of integral transforms: Fourier cosine and Laplace transforms. We note that from the above factorization equality, the general definition of convolutions has the form γ
∗(f1 , f2 , ..., fn )(x) = T −1 [γ (·)(T1 f1 )(·)(T2 f2 )(·)...(Tn fn )(·)] (x) with T −1 being the inverse operator of T . Although it looks quite simple, it is not easy to have an explicit form of convolutions when applied to concrete integral transforms. Furthermore, to obtain explicit formulas for convolutions of different integral transforms, one should answer the question in which function spaces the convolutions live and which properties they own. We will approach these goals for a new polyconvolution with a weight function for two Fourier cosine transforms and one Laplace transform. As a by-product, we will apply this new notion to solving some non-standard integral equations and systems of integral equations. We note that for such systems of integral equations, a representation of their solution in a closed form is an interesting and open problem [2, 7]. The paper is organized as follows. In Section 2, we recall some known
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