The convolution with a weight function related to the fourier cosine integral transform

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2010, Vol. 55, No. 6, pp. 90-100 THE CONVOLUTION WITH A WEIGHT FUNCTION RELATED TO THE FOURIER COSINE INTEGRAL TRANSFORM Nguyen Xuan Thao(∗) Hanoi University of Science and Technology Ta Duy Cong The Broadcasting and Television College I (∗)E-mail: thaonxfami@mail.hut.edu.vn Abstract. Convolutions with a class of weight function for the Fourier cosine integral transform in the spaces Lpα,β(R+) are studied. Existence conditions for these convolutions, a Young’s type theorem and a Parseval type equality are obtained. Applications to solve a class of the integral equations and systems of integral equations with function coefficiences are considered. Keywords: Convolution, weight function, Fourier transform. 1. Introduction The Fourier intergral transform of function f(x) is of the form (see [6, 9, 13, 14, 16]) (Ff)(y) = 1√ 2pi ∞∫ −∞ e−xy f(x) dx, (1.1) with Fourier convolution (see [13, 14]) ( f ∗ F g ) (x) = ∞∫ −∞ f(x− t)g(t)dt. (1.2) The Fourier cosine intergral transform of function f(x) was studied in [3, 4, 5, 13, 16] (Fcf)(y) = √ 2 pi ∞∫ 0 cos yx f(x) dx. (1.3) In 1951, Sneddon I. N. proposed the convolution of two functions f(x) and g(x) for the Fourier cosine integral transform (see [7, 13, 14]) ( f ∗ Fc g ) (x) = 1√ 2pi ∞∫ 0 f(y) [ g(x+ y) + g(|x− y|)] dy, (1.4) 90 The convolution with a weight function related to the Fourier cosine integral transform which satisfies the following factorization property Fc ( f ∗ Fc g ) (y) = (Fcf)(y) . (Fcg)(y) ∀y ∈ R+, (1.5) and the norm inequality ‖f ∗ Fc g‖ ≤ ‖f‖ . ‖ g ‖. (1.6) In 2004, Thao N. X. and Khoa N. M. studied the convolution with a weight function γ = cos y of two functions f(x) and g(x) for the Fourier cosine integral transform [15] ( f γ∗ g)(x) = 1 2 √ 2pi ∞∫ 0 f(t)[g(x+ 1 + t) + g(| x+ 1− t |) + g(| x− 1 + t |)+ + g(| x− 1− t) |)]dt, (1.7) and proved the factorization property Fc ( f γ∗ g)(y) = cos y (Fcf)(y).(Fcg)(y), ∀y ∈ R+, f, g ∈ L(R+). (1.8) Norm of the function f(x) in the space L(R+) is defined ‖f‖ = √ 2 pi ∞∫ 0 |f(x)| dx. (1.9) Then the convolution ( f γ∗ g)(x) belongs to the space L(R+) and satisfies the convolution inequality ‖f γ∗ g‖ ≤ ‖f‖ . ‖ g ‖. (1.10) Convolutions with a weight function in the space Lp(R) were studied [11, 12]. For instance, Young’s theorem and its corollary was proposed by himself. Theorem 1.1 (A Young’s theorem). [2] Let p, q, r > 1, 1 p + 1 q + 1 r = 2, and f ∈ Lp(R), g ∈ Lq(R), h ∈ Lr(R), then  ∞∫ 0 ( f ∗ F g ) (x) . h(x) dx  ≤ ‖f‖ Lp(R) ‖g‖ Lq(R) ‖h‖ Lr(R). (1.11) Corollary 1.1 (A Young’s inequality). [2] Let p, q, r > 1, 1 p + 1 q = 1 + 1 r and f ∈ Lp(R), g ∈ Lq(R), then (f ∗ F g ) (x) ∈ Lr(R) and ‖(f ∗ F g )‖Lr(R) ≤ ‖f‖Lp(R) ‖g‖Lq(R). (1.12) 91 Nguyen Xuan Thao and Ta Duy Cong In this paper, we study the convolution with a class of the weight function for the Fourier cosine integral transform in the spaces Lpα,β(R+). We obtained conditions for its existence, a Young’s type theorem and the Parseval type equality. Also, in application, we solved a class of the integral equations and systems of integral equation with function coefficients whose solutions can be presented in closed form. 2. The convolution and its properties in Lp(R+) Definition 2.1. The convolution with a weight function γ = cosαy, α ∈ R+ of two functions f(x) and g(x) for the Fourier cosine integral transform defined by ( f γ∗ g)(x) = 1 2 √ 2pi ∞∫ 0 f(t)[g(x+ α + t) + g(| x+ α− t |) + g(| x− α+ t |) + g(| x− α− t) |)]dt, x > 0. (2.1) Remark 2.1. If α = 0 then the convolution (2.1) coincides with the convolution studied in [4], if α = 1 then the convolution (2.1) coincides with the convolution studied in [15]. Similar arguments as in [15], we obtain following results Theorem 2.1. Let f(x), g(x) ∈ L(R+), then convolution (2.1) exists, belongs to L(R+) and satisfies the factorization property Fc ( f γ∗ g)(y) = cosαy (Fcf)(y).(Fcg)(y), ∀y ∈ R+. (2.2) Further, convolution operator (2.1) is continuous, bounded and satisfies Par- seval type equality ( f γ∗ g)(x) = √ 2 pi ∞∫ 0 ( Fcf ) (y). ( Fcg ) (y) cosαy cosxy dy. (2.3) Definition 2.2. The functional spaces Lp(R+) and L p α,βR+ are respestively defined by Lp(R+) = { f : ( ∞∫ 0 | f(x) |p dx ) 1 p < +∞ } ; (2.4) and Lpα,βR+ = { f : ( ∞∫ 0 tγe−βx | f(x) |p dx ) 1 p < +∞ } . (2.5) 92 The convolution with a weight function related to the Fourier cosine integral transform Theorem 2.2. (A Young’s type theorem) Let p, q, r > 1, such that 1 p + 1 q + 1 r = 2, and let f ∈ Lp(R+), g ∈ Lq(R+), h ∈ Lr(R+), then ∞∫ 0 ( f γ∗ g)(x) . h(x) dx ≤ √ 2 pi ‖f‖ Lp(R+) ‖g‖ Lq(R+) ‖h‖ Lr(R+). (2.6) Proof. In view of the convolution of two functions f(x) and g(x) with a weight function defined in (1.7), we have  ∞∫ 0 ( f γ∗ g)(x) . h(x) dx ≤ 1 2 √ 2pi ∫ ∞ 0 ∫ ∞ 0 |f(t)| |g(x+ α+ t) + g(|x+ α− t|)| |h(x)| dt dx + 1 2 √ 2pi ∞∫ 0 ∞∫ 0 |f(t)| |g(|x− α+ t|) + g(|x− α− t|)| |h(x)| dt dx. (2.7) Let p1, q1, r1 be the conjugate exponentials of p, q, r respestively, it means 1 p + 1 p1 = 1, 1 q + 1 q1 = 1, 1 r + 1 r1 = 1. It is obvious that 1 p1 + 1 q1 + 1 r1 = 1. Put U(x, y) = |g(x+ α + t) + g(|x+ α− t|)| qp1 . |h(x)| r p1 ; V (x, y) = |f(t)| p q1 . |h(x)| r q1 ; W (x, y) = |f(t)| pr1 . |g(x+ α + t) + g(|x+ α− t|)| qr1 . We have U(x, y) . V (x, y) .W (x, y) = |f(t)| . |h(x)| . |g(x+ α+ t) + g(|x+ α− t|)|. (2.8) Thus∫ ∞ 0 ∫ ∞ 0 |f(t)|. |g(x+ α + t) + g(|x+ α− t|)| |h(x)| dt dx = ∞∫ 0 ∞∫ 0 U(x, y) . V (x, y) .W (x, y)dtdx. 93 Nguyen Xuan Thao and Ta Duy Cong Applying the Ho¨lder’s inequality for three functions U(x, y) , V (x, y) , W (x, y), we have ∞∫ 0 ∞∫ 0 U(x, y) . V (x, y) .W (x, y)dydx ≤ ‖U‖Lp1(R2+) ‖V ‖Lq1(R2+) ‖W‖Lr1(R2+). It follows that∫ ∞ 0 ∫ ∞ 0 |f(t)| |g(x+ α+ t) + g(|x+ α− t|)| |h(x)| dt dx ≤ ‖U‖Lp1 (R2+) ‖V ‖Lq1(R2+) ‖W‖Lr1(R2+). (2.9) In the space Lp1(R2+), we have ‖U‖p1 Lp1 (R2+) = ∞∫ 0 ∞∫ 0 |g(x+ α + t) + g(|x+ α− t|)|q . |h(x)|r dx dt = ∞∫ 0 ( ∞∫ 0 |g(x+ α + t) + g(|x+ α− t|)|q dt ) |h(x)|r dx. (2.10) Note that |t|q is a convex function, hence ∞∫ 0 |g(x+ α + t) + g(| x+ α− t |)|q dt ≤ 2q−1 ( ∞∫ 0 |g(x+ α + t)|q dt+ ∞∫ 0 |g(| x+ α− t |)|q dt ) = 2q ∞∫ 0 | g(t) |q dt. (2.11) From (2.11) and (2.10), we get wwUwwp1 Lp1(R2+) ≤ 2q ∞∫ 0 | g(t) |q dt ∞∫ 0 |h(x)|r dx = 2q ‖g‖qLq(R+) ‖h‖rLr(R+). (2.12) Similarly, in the space Lr1(R2+), we havewwWwwr1 Lr1(R2+) ≤ 2q ‖f‖pLp(R+) ‖g‖ q Lq(R+) . (2.13) In the space Lq1(R2+), we havewwVww L q1(R2+) = ‖f‖ p q1 Lp(R+) ‖h‖ r q1 Lr(R+) . (2.14) 94 The convolution with a weight function related to the Fourier cosine integral transform From formulas (2.12), (2.13) and (2.14), we havewwUww Lp1 (R2+) wwVww Lq1(R2+) wwWww Lr1(R2+) ≤ 2. ‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+). (2.15) From (2.15) and (2.9), we have ∣∣∣∣ ∞∫ 0 ∞∫ 0 f(t)[g(x+ α+ t) + g(| x+ α− t |)] h(x) dt dx ∣∣∣∣ ≤ 2. ‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+). (2.16) Similarly ∞∫ 0 ∞∫ 0 |f(t)| |g(|x− α + t|) + g(|x− α− t|)| h(x) dt dx ≤ 2. ‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+). (2.17) From formulas (2.16), (2.17) and (2.7), we get ∞∫ 0 ( f γ∗ g)(x) . h(x) dx ≤ √ 2 pi ‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+). The proof is complete. Corollary 2.1. Let p, q, r > 1 such that 1 p + 1 q = 1 + 1 r , and let f ∈ Lp(R+), g ∈ Lq(R+), then ( f γ∗ g)(x) ∈ Lr(R+) and ‖(f γ∗ g)‖Lr(R+) ≤ √ 2 pi ‖f‖Lp(R+) ‖g‖Lq(R+). (2.18) Theorem 2.3. Let f ∈ Lp(R+), g ∈ Lq(R+), 0 1 such that 1 p + 1 q = 1. Then the convolution ( 2.1) exists, is continuous and bounded. Further- more, ( f γ∗ g) ∈ Lrα,γ(R+) and the following estimation holds ‖(f γ∗ g)‖Lrα,γ(R+) ≤ 2q√ 2pi C ‖f‖Lp(R+) ‖g‖Lq(R+), (2.19) where C = β− γ+1 r .Γ 1 r (γ + 1), γ > −1. Further, if f ∈ Lp(R+) ∩ L(R+), g ∈ Lq(R+) ∩ L(R+), the convolution (2.1) satisfies the factorization property (2.2), and the Parseval’s type equality (2.3) holds. This theorem can be proved easily basing on the Ho¨lder’s inequality for con- volution (2.1), formula (3.225.3) in [10] and Riemann - Lebesgue lemma. 95 Nguyen Xuan Thao and Ta Duy Cong 3. Applications 3.1. Integral equations with function coefficient Consider the integral equation f(x) + λ(x) 2 √ 2pi ∞∫ 0 f(t)ψ(g)(x, t) dt = h(x). (3.1) Here, g, h are two given continuous functions in L(R+), λ is a continuous function such that 0 < m ≤ |λ(x)|, f(x) is an unkown function and ψ(g)(x, t) = 1 λ(t) [ g(x+α+ t)+g(|x+α− t|)+g(|x+α− t|)+g(|x−α− t|)]. (3.2) Theorem 3.1. Suppose that 1+cosαy ( Fcg ) (y) 6= 0, ∀ y ∈ R+, then there exists the unique solution in L(R+) of equation (3.1), namely, a closed form of this solution is of the form f(x) = h(x)− (h λ γ∗ ϕ)(x).λ(x). (3.3) where ϕ(x) is a continuous function, belongs to L(R+) defined by ( Fcϕ ) (y) = ( Fcg ) (y) 1 + cosαy ( Fcg ) (y) . Proof. We can rewrite equation (3.1) as follows f(x) + λ(x) (f λ γ∗ g)(x) = h(x). (3.4) The given condition shows that λ(x) 6= 0 for all x ∈ R+, hence the equation (3.4) is equivalent to f(x) λ(x) + (f λ γ∗ g)(x) = h(x) λ(x) . (3.5) Applying the Fourier cosine integral transform to both sides of (3.5) and using the factorization equality (2.2), we get ( Fc f λ ) (y) [ 1 + cosαy ( Fcg ) (y) ] = ( Fc h λ ) (y). By virtue of the hypothesis, we have ( Fc f λ ) (y) = ( Fc h λ ) (y) 1 + cosαy ( Fcg ) (y) , 96 The convolution with a weight function related to the Fourier cosine integral transform or equivalently, ( Fc f λ ) (y) = ( Fc h λ ) (y) [ 1− cosαy ( Fcg ) (y) 1 + cosαy ( Fcg ) (y) ] . (3.6) Due to Wiener-Levy’s theorem [1], there exists a unique function ϕ(x) ∈ L(R+) such that ( Fcϕ ) (y) = ( Fcg ) (y) 1 + cos y ( Fcg ) (y) . Therefore equation (3.6) becomes ( Fc f λ ) (y) = ( Fc h λ ) (y) [ 1− cosαy(Fcϕ)(y)] = ( Fc h λ ) (y)− Fc (h λ γ∗ ϕ)(y) (3.7) Applying the inverse Fourier cosine integral transform to both sides of equation (3.7), we get f(x) λ(x) = h(x) λ(x) − (h λ γ∗ ϕ)(x). It follows the solution of equation (3.1) f(x) = h(x)− (h λ γ∗ ϕ)(x).λ(x). Note that conditions m < |λ(x)| and h ∈ L(R+) imply that h λ ∈ L(R+). By Theorem 2.1, we get f(x) ∈ L(R+). The theorem is proved completely. 3.2. The integral equation system with function coefficient Finally, we consider the following system of integral equations  f(x) + λ1(x) 2 √ 2pi ∞∫ 0 g(t) h(ϕ)(x, t)dt = p (x) λ2(x) 2 √ 2pi ∞∫ 0 f(t) k(ψ)(x, t)dt+ g(x) = q(x), (3.8) where, ϕ, ψ, p, q are continuous functions, belong to L(R+), λ1, λ2 are given such that 0 < m ≤ |λj(x)|, j = 1, 2 for some positive m, and f , g are unkown functions and h(ϕ)(x, t) = 1 λ2(t) [ ϕ(x+α+t)+ϕ(| x+α−t |)+ϕ(| x−α+t |)+ϕ(| x−α−t |)], (3.9) 97 Nguyen Xuan Thao and Ta Duy Cong k(ψ)(x, t) = 1 λ1(t) [ ψ(x+α+ t)+ψ(| x+α− t |)+ψ(| x−α+ t |)+ψ(| x−α− t |)]. (3.10) Theorem 3.2. Suppose that the condition 1− cosαy Fc ( ϕ γ∗ ψ)(y) 6= 0 holds for all y > 0. Then the system (3.8) has a unique solution (f, g) ∈ L(R+)× L(R+) defined as follows f(x) = p(x)− λ1(x). ( q λ2 γ∗ ϕ)(x) + λ1(x).[( p λ1 − ( q λ2 γ∗ ϕ)) γ∗ ξ](x), g(x) = q(x)− λ2(x). ( p λ1 γ∗ ψ)(x) + λ2(x).[( q λ2 − ( p λ1 γ∗ ψ)) γ∗ ξ](x), (3.11) Here, ξ(x) is a unique continuous function belonging to L(R+) and defined by ( Fc ξ ) (y) = Fc ( ϕ γ∗ ψ)(y) 1− cosαy Fc ( ϕ γ∗ ψ)(y) . (3.12) Proof. The equation system (3.8) can be rewritten as follows  f(x) + λ1(x). ( g λ2 γ∗ ϕ)(x) = p (x) λ2(x). ( f λ1 γ∗ ψ)(x) + g(x) = q(x). (3.13) Applying Fourier cosine transform Fc to both sides of each equations of (3.13), we obtain   Fc ( f λ1 ) (y) + cosαy Fc ( g λ2 ) (y) ( Fcϕ ) (y) = Fc ( p λ1 ) (y) cosαy Fc ( f λ1 ) (y) ( Fcψ ) (y) + Fc ( g λ2 ) (y) = Fc ( q λ2 ) (y). (3.14) Under the condition of the hypothesis, it follows Cramer linear equation system (3.14) having a unique solution  Fc ( f λ1 ) (y) = Fc ( p λ1 − ( q λ2 γ∗ ϕ))(y) 1− cosαy Fc ( ϕ γ∗ ψ)(y) Fc ( g λ2 ) (y) = Fc ( q λ2 − ( p λ1 γ∗ ψ))(y) 1− cosαy Fc ( ϕ γ∗ ψ)(y) , or equivalently,  Fc ( f λ1 ) (y) = Fc ( p λ1 − ( q λ2 γ∗ ϕ))(y).[1 + cosαy Fc ( ϕ γ∗ ψ)(y) 1− cosαy Fc ( ϕ γ∗ ψ)(y) ] Fc ( g λ2 ) (y) = Fc ( q λ2 − ( p λ1 γ∗ ψ))(y).[1 + cosαy Fc ( ϕ γ∗ ψ)(y) 1− cosαy Fc ( ϕ γ∗ ψ)(y) ] . 98 The convolution with a weight function related to the Fourier cosine integral transform Due to Wiener-Levy’s Theorem [1], there exists a unique continuous function ξ ∈ L(R+) such that ( Fc ξ ) (y) = Fc ( ϕ γ∗ ψ)(y) 1− cosαy Fc ( ϕ γ∗ ψ)(y) . Therefore  Fc ( f λ1 ) (y) = Fc [ p λ1 − ( q λ2 γ∗ ϕ)](y).[1 + cosαy (Fc ξ)(y)] Fc ( g λ2 ) (y) = Fc [ q λ2 − ( p λ1 γ∗ ψ)](y).[1 + cosαy (Fc ξ)(y)]. We obtain  Fc ( f λ1 ) (y) = Fc [ p λ1 − ( q λ2 γ∗ ϕ)](y) + Fc[( p λ1 − ( q λ2 γ∗ ϕ)) γ∗ ξ](y) Fc ( g λ2 ) (y) = Fc [ q λ2 − ( p λ1 γ∗ ψ)](y) + Fc[( q λ2 − ( p λ1 γ∗ ψ)) γ∗ ξ](y). It shows that  f(x) λ1(x) = [ p λ1 − ( q λ2 γ∗ ϕ)](x) + [( p λ1 − ( q λ2 γ∗ ϕ)) γ∗ ξ](x) g(x) λ2(x) = [ q λ2 − ( p λ1 γ∗ ψ)](x) + [( q λ2 − ( p λ1 γ∗ ψ)) γ∗ ξ](x). Therefore, we obtain a unique solution of system (3.8)  f(x) = p(x)− λ1(x). ( q λ2 γ∗ ϕ)(x) + λ1(x).[( p λ1 − ( q λ2 γ∗ ϕ)) γ∗ ξ](x) g(x) = q(x)− λ2(x). ( p λ1 γ∗ ψ)(x) + λ2(x).[( q λ2 − ( p λ1 γ∗ ψ)) γ∗ ξ](x) Conditions 0 < m ≤ |λj(x)|, j = 1, 2 and p, q ∈ L(R+) imply that pλ1 , q λ2 ∈ L(R+). By Theorem 2.1, it is obvious that f(x) and g(x) belong to L(R+). The proof is complete. Remark 3.1. One can easily extend above results to weight functions cosαx, α ∈ R. REFERENCES [1] Achiezer N. I., 1965. Lectures on Approximation Theory. Science Publishing House, Moscow. [2] Adams R. A. and Fourier J. J. S., 2003. Sobolev Spaces, 2nd ed. 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