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

factorization property Fc γ ∗ f, g, h y siny 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.


Introduction
The convolution of two functions f and g for the Fourier transform is well known 1 The convolution of f and g for the Kontorovich-Lebedev integral transform has been studied in 2 xu v xv u uv x f u g v du dv, x > 0, 1.4 for which the following factorization identity holds: Here K iy is the Kontorovich-Lebedev transform 3 K ix t f t dt, 1.6 and K ix t is the Macdonald function 4 . The convolution of two functions f and g for the Fourier cosine is of the form 1 f y g x − y g x y dy, x > 0, 1.7 which satisfied the following factorization equality: g y F c f y F c g y , ∀y > 0. 1.8 Here the Fourier cosine transform is of the form cos yx · f x dx, y > 0. 1.9 The convolution with a weight function γ x sin x of two functions f and g for the Fourier sine transform has been introduced in 5, 6 Here the Fourier sine is of the form The generalized convolution of two functions f and g 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 f and g 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 γ x sin x for the Fourier cosine and the Fourier sine transforms of f and g has been introduced in 8 The generalized convolution with the weight function γ x sin x of f and g 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. 21 The respective factorization equalities are 10 In 1997, Kakichev introduced a constructive method for defining a polyconvolution with a weight function γ of functions f 1 , f 2 , . . . , f n for the integral transforms K, K 1 , K 2 , . . . , K n , which are denoted by γ * f 1 , f 2 , . . . , f n x , 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 f, g, and h for the Fourier cosine and the Fourier sine transforms has the form 13 * f, g, h x 1 2π Mathematical Problems in Engineering 5 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 k 1 , k 2 , and g are known functions, and f 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.

Polyconvolution
Definition 2.1. The polyconvolution with the weight function γ sin x of functions f, g, and h for the Fourier cosine, Fourier sine, and the Kontorovich-Lebedev integral transforms is defined as follows:

2.2
Theorem 2.2. Let f and g be functions in L 1 R , and let h be a function in L 1 1/ √ w, R ; then the polyconvolution 2.1 belongs to L 1 R and satisfies the following factorization equality: Mathematical Problems in Engineering

2.4
On the other hand, note that cosh

2.6
It shows that

2.7
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 L 1 R . We now prove the factorization equality 2.3 . Indeed, we have sin y F s f y F c g y K iy h 2 π ∞ 0 sin y sin yu cos yv K iy w f u g v h w du dv dw.

2.9
Mathematical Problems in Engineering 7 Using formula 2, page 130 in 4 , we get sin y F s f y F c g y K iy h 2 π ∞ 0 sin y sin yu cos yv cos yα e −w cosh α f u g v h w du dv dw dα Interchanging variables, we have

2.12
From fomulae 2.10 -2.8 , we have sin y F s f y F c g y K iy h F c γ * f, g, h y .

2.13
The proof is complete.

Mathematical Problems in Engineering
Definition 2.3. Let f be a function in L 1 R and let h be a function in L 1 β, R ; their norms are defined as follows: Theorem 2.4. Let f and g be functions in L 1 R , and let h be function in L 1 β, R ; then the following estimation holds: Proof. From formulas 2.1 , 2.2 , and 2.7 , we have Therefore, by Definition 2.3,

2.17
Proposition 2.5. Let f, g ∈ L 1 R , and let h ∈ L 1 1/ √ w, R ; then the following identity holds:

2.18
Proof. From the definition 2.1 of the polyconvolution and the convolution 1.7 , we have

2.19
Mathematical Problems in Engineering 9 From 2.19 and calculation, we obtain *

2.20
The proof is complete.
Theorem 2.6. Let f, g, h be functions in L 1 R , γ x sin x, and let l and k be functions in L 1/ √ w, R ; then the following properties holds: Proof. First, we prove the assertion c . h, k .

2.21
Therefore, the part c holds. Other parts can be proved in a similar way.

Applications in Solving Integral Equations and Systems of Integral Equations
Consider the integral equation where g, h, k, l, and ϕ are known functions, f is an unknown function, θ x, u, v, w is given by the formula 2.2 , and Here ξ ∈ L 1 R is defined uniquely by F c ξ y sin y F s g y K iy h sin y F s k 1 y F c k 2 y F c l y 1 sin y F s g y K iy h sin y F s k 1 y F c k 2 y F c l y .

3.5
Proof. We obtain the following lemmas.

Lemma 3.2.
For f, k ∈ L 1 R , then the following operator also belongs to L 1 R ∞ 0 f u θ 1 x, u du.

3.6
Moreover, the following factorization equality holds: to L 1 R and the respectively factorization equality is h y sin y F s g y K iy h , ∀y > 0, 3.9 We now prove Theorem 3.1 with the help of convolution 1.7 , Lemmas 1, and 2. We have F c f y sin y F s g y · F c f y · K iy h F s k y · F c f y sin y F c l y · F c f y F c ϕ y .

3.10
Therefore, by the given condition, The proof is complete.

3.16
Next, we consider the following system of two integral equations:

3.17
Here θ x, u, v, w is defined by 2.2 , and

3.18
h, k, l, ξ, η, p, q are known functions, and f and g are unknown functions.
Theorem 3.5. Given that p, q, h, l, ξ, η 1 , η 2 ∈ L 1 R and k ∈ L 1 β, R , η η 1 * 3 η 2 such that Mathematical Problems in Engineering 13 Then the system 3.17 has a unique solution in L 1 R × L 1 R whose closed form is as follows

3.20
Here, l ∈ L 1 R is defined by Proof. We need the following lemma.

3.28
It shows that