Abstract
The polyconvolution with the weight function of three functions , and for the integral transforms Fourier sine , Fourier cosine , and Kontorovich-Lebedev , which is denoted by (x), has been constructed. This polyconvolution satisfies the following factorization property , for all . 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.
1. Introduction
The convolution of two functions and for the Fourier transform is well known [1] This convolution has the factorization equality as below where denotes the Fourier transform The convolution of and for the Kontorovich-Lebedev integral transform has been studied in [2] for which the following factorization identity holds: Here is the Kontorovich-Lebedev transform [3] and is the Macdonald function [4].
The convolution of two functions and for the Fourier cosine is of the form [1] which satisfied the following factorization equality: Here the Fourier cosine transform is of the form The convolution with a weight function of two functions and for the Fourier sine transform has been introduced in [5, 6] and the following factorization identity holds: Here the Fourier sine is of the form The generalized convolution of two functions and for the Fourier sine and Fourier cosine transforms has been studied in [1] and proved the following factorization identity [1]: The generalized convolution of two functions and for the Fourier cosine and the Fourier sine transforms is defined by [7] For this generalized convolution, the following factorization equality holds: The generalized convolution with the weight function for the Fourier cosine and the Fourier sine transforms of and has been introduced in [8] It satisfies the factorization property The generalized convolution with the weight function of and for the Fourier sine and Fourier cosine has been studied in [9] and satisfies the following factorization identity: Recently, the following generalized convolutions for Fourier cosine, Kontorovich-Lebedev and Fourier sine, Kontorovich-Lebedev are studied in [10] (f. ) The respective factorization equalities are [10] where
In 1997, Kakichev introduced a constructive method for defining a polyconvolution with a weight function of functions for the integral transforms , which are denoted by , such that the following factorization property holds [11]: Polyconvolutions for the Hilbert, Stieltjes, Fourier cosine, and Fourier sine integral transforms have been studied in [12].
The polyconvolution of , , and for the Fourier cosine and the Fourier sine transforms has the form [13] which satisfies the following factorization property: In recent years, many sciences were interested in the theory of convolution for the integral transforms and gave several interesting application (see [3, 14–21]), specially, the integral equations with the Toeplitz plus Hankel kernel [22–24] where and are known functions, and is an unknown function. Many partial cases of this equation can be solved in closed form with the help of the convolutions and generalized convolutions. In this paper, we construct and investigate the polyconvolution for the Fourier sine, Fourier cosine, and the Kontorovich-Lebedev transforms. Several properties of this new polyconvolution and its application on solving integral equation with Toeplitz plus Hankel equation and systems of integral equations are obtained.
2. Polyconvolution
Definition 2.1. The polyconvolution with the weight function of functions , , and for the Fourier cosine, Fourier sine, and the Kontorovich-Lebedev integral transforms is defined as follows: where
Theorem 2.2. Let and be functions in , and let be a function in ; then the polyconvolution (2.1) belongs to and satisfies the following factorization equality:
Proof. Since for sufficient large , we have On the other hand, note that ; we have Using formula , page 321, in [4], we have It shows that By the same way, we obtain similar estimations for the 7 other terms. Therefore, from formulas (2.1), (2.2), and (2.7), we have It shows that the polyconvolution (2.1) belongs to . We now prove the factorization equality (2.3). Indeed, we have Using formula , page 130 in [4], we get Interchanging variables, we have Similarly, From fomulae (2.10)–( 2.8), we have The proof is complete.
Definition 2.3. Let be a function in and let be a function in ; their norms are defined as follows: here .
Theorem 2.4. Let and be functions in , and let be function in ; then the following estimation holds:
Proof. From formulas (2.1), (2.2), and (2.7), we have Therefore, by Definition 2.3,
Proposition 2.5. Let , and let ; then the following identity holds:
Proof. From the definition (2.1) of the polyconvolution and the convolution (1.7), we have From (2.19) and calculation, we obtain The proof is complete.
Theorem 2.6. Let be functions in , and let and be functions in ; then the following properties holds: (a)(b)(c)(d)(e)
Proof. First, we prove the assertion (c). From Theorem 2.2 and the convolutions (1.17), (1.10), we have Therefore, the part (c) holds. Other parts can be proved in a similar way.
3. Applications in Solving Integral Equations and Systems of Integral Equations
Consider the integral equation where and are known functions, is an unknown function, is given by the formula (2.2), and
Theorem 3.1. Suppose that , , such that then (3.1) has a unique solution in whose closed form is Here is defined uniquely by
Proof. We obtain the following lemmas.Lemma 3.2. For , then the following operator also belongs to Moreover, the following factorization equality holds: Lemma 3.3. Let , ; then the generalized convolution belongs to and the respectively factorization equality is where We now prove Theorem 3.1 with the help of convolution (1.7), Lemmas 1, and 2. We have Therefore, by the given condition, By the hypothesis , we see that ; using Lemma 3.3, we get In virtue of the Wiener-Levy theorem [25], by the given condition, there exists a function such that From (3.12) and (3.13), we have Then the solution in of (3.1) has the form The proof is complete.
Remark 3.4. The integral equation (3.1) is a special case of the integral equation with the Toeplitz plus Hankel kernel (1.27) for and Next, we consider the following system of two integral equations: Here is defined by (2.2), and are known functions, and and are unknown functions.
Theorem 3.5. Given that and , such that , where Then the system (3.17) has a unique solution in whose closed form is as follows Here, is defined by
Proof. We need the following lemma.Lemma 3.6. Let ; then Using Theorem 2.2, Lemma 3.6, and the generalized convolution (1.15), (1.19), we have On the other hand, from we have . Therefore, using Theorem 2.2 and the generalized convolution (1.15), (1.17), we have Hence, in view of the Wiener-Levy theorem [25], by the given condition, there is a unique function such that where On the other hand, using Theorem 2.2 and the generalized convolution (1.15), we have Hence, from (3.25), (3.27) we have It shows that Similarly, from the generalized convolutions (1.15), (1.19), we have Using formulas (3.25), (3.30), we have It shows that Pair defined by fomulae (3.29) and (3.32) is a solution in closed form in of system (3.17). The proof is complete.
Acknowledgments
In the memory of professor V. A. Kakichev, the author wish to express his deep thanks to him for all his encouragement to the author in this investigative direction. This research is supported partially by Vietnam’s National Foundation for Science and Technology Development, Grant no. 101.01.21.09.