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.
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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, T 1, T 2, … , T n with the weight function γ(x) of functions f 1, f 2, … , f n for which the factorization property holds
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 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
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 convolutions and generalized convolutions. In Section 3, we define a new polyconvolution with a weight function γ(y) = e −y of three functions for Fourier cosine and Laplace transforms and prove the existence of this polyconvolution in certain function spaces as well as the factorization equality and algebraic properties of this polyconvolution operator. In Section 4, the boundedness property of the polyconvolution operator is considered. In Section 5, we study an integral transform related to this polyconvolution. Finally, in Section 6 with the help of the new polyconvolution, we study a class of Toeplitz plus Hankel integral equations and some systems of integral equations and prove that they can be solved in a closed form.
2 Preliminaries
In this section, we recall some known convolutions and generalized convolutions. The convolution of two functions f and g in \(L_{1}(\mathbb {R})\) for the Fourier integral transform is well known [8] as
for which the following factorization property holds:
where the Fourier integral transform is defined by
The convolution of two functions f and g for the Laplace transform is that of the form [8]
which satisfies the factorization identity
Here L denotes the Laplace integral transform [8]
The convolution of two functions f and g in \(L_{1}(\mathbb {R}_{+})\) for the Fourier cosine transform is defined by [16]
with the factorization equality
Here, the Fourier cosine integral transform is defined by [16]
The generalized convolution for the Fourier sine and Fourier cosine transforms is defined by [8]
for the which the factorization equality holds:
where F s denotes the Fourier sine integral transform [8]
The generalized convolution for the Fourier cosine and Fourier sine transforms is defined by [12]
which satisfies the factorization identity
The convolution with the weight function γ 1(y) = cosy for the Fourier cosine transforms is defined by [9]
which satisfies the factorization identity
3 The Polyconvolution with the Weight Function γ(y) = e −y for Fourier Cosine and Laplace Transforms
Definition 3.1
The polyconvolution with the weight function γ(y) = e −y of three functions f, g, h for Fourier cosine and Laplace transforms is defined by
Theorem 3.2
If the functions f, g, h are given in \(L_{1}(\mathbb {R}_{+})\) , then the polyconvolution (13) belongs to \(L_{1}(\mathbb {R}_{+})\) and satisfies the factorization identity
Moreover, when \(f,g,h\in L_{2}(\mathbb {R}_{+})\cap L_{1}(\mathbb {R}_{+})\), the Parseval type identity holds:
Proof
We have
Therefore,
Thus
On the other hand, using (13) and the formula \( {\int }_{0}^{\infty } {{{e}^{-sx}}\cos xydx}=\frac {s}{s^{2}+{{y}^{2}}},\ s>0 \), we obtain
Thus the Parseval type identity (15) holds. Combining with (17), we get the factorization identity (14). The theorem is proved. □
Proposition 3.3
Let \(f,g,h,l\in L_{1}(\mathbb {R}_{+})\). Then the polyconvolution (13) satisfies the following equalities
-
(a)
\( \overset {\gamma } {{*}}\,(f,g,h) = \overset {\gamma } {{*}}\,(g,f,h)\),
-
(b)
\( \overset {\gamma } {{*}}\,\left [ \left (f\underset {F_{c}}{{*}}\,g \right ),l,h \right ]=\overset {\gamma } {{*}}\,\left [ \left (f\underset {F_{c}}{{*}}\,l \right ),g,h \right ]\),
-
(c)
\( \overset {\gamma } {{*}}\,\left [ \left (f\underset {F_{c}}{{*}}\,g \right ),l,h \right ]=\overset {\gamma } {{*}}\,\left [ \left (l\underset {F_{c}}{{*}}\,g \right ),f,h \right ]\).
Proof
First, we show (a). Indeed, from the factorization equality (14), we have
Thus \( \overset {\gamma } {{*}}\,(f,g,h) = \overset {\gamma } {{*}}\,(g,f,h)\).
(b) Using the factorization properties (6) and (14), we can write
So, we get \( \overset {\gamma } {{*}}\,[ (f\underset {F_{c}}{{*}}\,g ),l,h ]=\overset {\gamma } {{*}}\,[ (f\underset {F_{c}}{{*}}\,l ),g,h ]\).
Similarly, we can prove (c). □
Theorem 3.4
(Tichmarch type theorem) Let \(f\in L_{1}(\mathbb {R}_{+}, e^{\alpha x}), \alpha >0, g, h \in L_{1}(\mathbb {R}_{+})\). If \(\overset {\gamma } {{*}}\,\left (f,g,h \right )\left (x \right ) = 0,\ x>0\), then either f(x) = 0, x > 0 or g(x) = 0, x > 0 or h(x) = 0, x > 0.
Proof
We have
Here, we used the estimation
Since \(f \in L_{1}(\mathbb {R}_{+}, e^{\alpha x})\), we get \(\frac {{{d}^{n}}}{d{{y}^{n}}}\left [ \cos yx f\left (x \right )\right ]\in {{L}_{1}}\left ({{\mathbb {R}}_{+}} \right ).\)
Since \({{L}_{1}}\left ({{\mathbb {R}}_{+}},{{e}^{\alpha x}} \right )\subset {{L}_{1}}\left ({{\mathbb {R}}_{+}} \right ),\left ({{F}_{c}}f \right )\left (y \right )\) is analytic in \(\mathbb {R}_{+}\). Similarly we obtain (F c g) is analytic in \(\mathbb {R}_{+}\). On the other hand, (L h)(y) is analytic in \(\mathbb {R}_{+}\). By using the factorization property (14) for \(\overset {\gamma } {{*}}\,\left (f,g,h \right )\left (x \right ) = 0\), we have
It implies that, either f(x) = 0, x > 0 or g(x) = 0, x > 0 or h(x) = 0, x > 0.
The theorem is proved. □
4 Inequalities for the Polyconvolution
In this section, we present the norm inequalities for the polyconvolution (13) in \(L_{1}(\mathbb {R}_{+})\) and \(L_{p}(\mathbb {R}_{+},\rho )\) with 1≤p ≤ ∞ and ρ being a weight function. The standard norms are defined as follows
Theorem 4.1
If f, g, h belong to \(L_{1}(\mathbb {R}_{+})\) , then the following inequality holds
Proof
From Definition 3.1 and the proof of Theorem 3.2, we obtain
So, we obtain (18). □
Next, we study the polyconvolution on the function space \(L_{s}(\mathbb {R}_{+}, e^{-\alpha x})\) and estimate its norm.
Theorem 4.2
Let \(f\in L_{p}(\mathbb {R}_{+}),\ g\in L_{q}(\mathbb {R}_{+}),\ h\in L_{r}(R_{+})\), be such that p, q, r > 1 and \(\frac {1}{p}+\frac {1}{q}+\frac {1}{r}=2\) . Then the polyconvolution (13) is bounded in \(L_{s}(\mathbb {R}_{+}, e^{-\alpha x})\) when s > 1, α > 0 and the following estimation holds
Proof
From the proof of Theorem 3.2, we have the following estimation
Let p 1, q 1, r 1 be the conjugate exponentials of p, q, r and
We see that U V W = |f(u)||g(v)||h(y)|.
Using Fubini’s theorem, we get
Similarly, we get
Since \(\frac {1}{p}+\frac {1}{q}+\frac {1}{r}=2\), we have \(\frac {1}{{{p}_{1}}}+\frac {1}{{{q}_{1}}}+\frac {1}{{{r}_{1}}}=1\). Using Hölder’s inequality and (20), we obtain
Since \({\int }_{0}^{\infty } {{{e}^{-\alpha x}}dx}=\frac {1}{\alpha } ,\ \alpha >0\), we have
So, we obtain
Thus, we have (19). The theorem is proved. □
5 The Integral Transform Related to this Polyconvolution
Now, we study an integral transform related to the polyconvolution (13), namely the transforms of the form
Similarly to [15], we can prove the following result.
Theorem 5.1
(Watson type theorem) Suppose that \(f, k_{1},\ k_{2}\in L_{2}(\mathbb {R}_{+})\cap L_{1}(\mathbb {R}_{+})\) are given functions. Then the condition
is a necessary and sufficient condition for the operator \(T_{k_{1},k_{2}}\) to be unitary on \(L_{2}(\mathbb {R}_{+})\) . Moreover, the inverse operator of \(T_{k_{1},k_{2}}\) takes the form
where \(\bar k_{1},\ \bar k_{2}\) are the complex conjugate functions of k 1 ,k 2 respectively. So, we obtain
6 Integral Equations and Systems of Equations
The polyconvolution (13) allows us to obtain the solutions for integral equations and systems of integral equations in closed form.
6.1 Consider the Following Integral Equation
where λ is a complex constant; g, h, k are functions in \(L_{1}(\mathbb {R}_{+}); and f(x)\) is an unknown function in \(L_{1}(\mathbb {R}_{+})\).
Theorem 6.1
Assume that 1+λe −y (F c g)(y)(Lh)(y) ≠ 0 ∀y > 0. Then the integral (24) has a unique solution in \(L_{1}(\mathbb {R}_{+})\) in the form
where \(l\in L_{1}(\mathbb {R}_{+})\) is defined by
Proof
Using Definition 3.1, (24) can be rewritten in the form
Because f and \( \overset {\gamma } {{*}}\ \left (f,g,h \right )\left (x \right ) \) and k are functions in \(L_{1}(\mathbb {R}_{+})\), one can apply the factorization property (14) to get
Thus
With the condition 1+λ e −y(F c g)(y)(L h)(y) ≠ 0 ∀y > 0, we have
Due to the Wiener-Lévy theorem [1], there exists a function \(l\in L_{1}(\mathbb {R}_{+})\) such that
From (25), (26), and (8), we obtain
Therefore
The proof is complete. □
6.2 Consider the System of Two Integral Equations
where λ 1, λ 2 are complex constants, φ 1, φ 2, φ 3, p, q are functions in \(L_{1}(\mathbb {R}_{+}), and f, g\) are unknown functions.
Theorem 6.2
If the following condition is satisfied
then there exists a unique solution in \(L_{1}(\mathbb {R}_{+})\) of system (27), which is defined by
Proof
Using Definition 3.1 and (9), the system (27) can be rewritten in the form
Due to the factorization properties (6), (14) we obtain the linear system of algebraic equations
The inverse of the determinant of this system has the form
According to the Wiener-Lévy theorem [1], there exists a function \(l\in L_{1}(\mathbb {R}_{+})\) such that
Hence
Therefore, using (6), we have
It follows that
Similarly, we obtain the formula for g as stated in the theorem. □
6.3 A System of Three Integral Equations
Here, λ 1, λ 2, λ 3 are complex constants; φ 1, φ 2, φ 3, φ 4, p, q, r are functions from \( L_{1}(\mathbb {R}_{+})\); and f, g, h are unknown functions.
Theorem 6.3 Under the condition
there exists a unique solution in \(L_{1}(\mathbb {R}_{+})\) of (29) given by
Proof
The system (29) can be rewritten in the form
Using the factorization identities (6), (11), and (14), we obtain
The determinant of this system is
Thus,
According to the Wiener-Lévy’s theorem ([1]), there exists a function l∈L 1(ℝ) such that
So, we obtain
From this we arrive at (30). □
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Khoa, N.M. On a Polyconvolution with a Weight Function for Fourier Cosine and Laplace Transforms. Acta Math Vietnam 41, 549–562 (2016). https://doi.org/10.1007/s40306-015-0154-8
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DOI: https://doi.org/10.1007/s40306-015-0154-8
Keywords
- Laplace transform
- Fourier cosine transform
- Convolution
- Generalized convolution
- Polyconvolution
- Integral equations