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

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Convolutions with a class of weight function for the Fourier cosine integral transform in the spaces Lp a,B(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.

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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]) Z∞ 1 (Ff )(y) = √ e−xy f (x) dx, (1.1) 2π −∞ with Fourier convolution (see [13, 14]) Z∞ f ∗ g (x) = f (x − t)g(t)dt. (1.2) F −∞ The Fourier cosine intergral transform of function f (x) was studied in [3, 4, 5, 13, 16] r Z∞ 2 (Fc f )(y) = cos yx f (x) dx. (1.3) π 0 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]) Z∞ 1 f ∗ g (x) = √ f (y) g(x + y) + g(|x − y|) dy, (1.4) Fc 2π 0 90
The convolution with a weight function related to the Fourier cosine integral transform which satisfies the following factorization property Fc f ∗ g (y) = (Fc f )(y) . (Fcg)(y) ∀y ∈ R+ , (1.5) Fc and the norm inequality kf ∗ gk ≤ kf k . k g k. (1.6) Fc 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] Z∞ γ 1 f ∗ g (x) = √ f (t)[g(x + 1 + t) + g(| x + 1 − t |) + g(| x − 1 + t |)+ 2 2π 0 + g(| x − 1 − t) |)]dt, (1.7) and proved the factorization property γ ∀y ∈ R+ , f, g ∈ L(R+ ). Fc f ∗ g (y) = cos y Fc f (y). Fc g (y), (1.8) Norm of the function f (x) in the space L(R+ ) is defined r Z∞ 2 kf k = |f (x)| dx. (1.9) π 0 γ Then the convolution f ∗ g (x) belongs to the space L(R+ ) and satisfies the convolution inequality γ kf ∗ gk ≤ kf k . k g k. (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. 1 1 1 Theorem 1.1 (A Young’s theorem). [2] Let p, q, r > 1, + + = 2, and f ∈ p q r Lp (R), g ∈ Lq (R), h ∈ Lr (R), then  Z∞  f ∗ g (x) . h(x) dx ≤ kf k Lp (R) kgk Lq (R) khk Lr (R) . (1.11)    F 0 1 1 1 Corollary 1.1 (A Young’s inequality). [2] Let p, q, r > 1, + = 1 + and f ∈ p q r Lp (R), g ∈ Lq (R), then f ∗ g (x) ∈ Lr (R) and F k f ∗ g kLr (R) ≤ kf kLp (R) kgkLq (R) . (1.12) F 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 = cos αy, α ∈ R+ of two functions f (x) and g(x) for the Fourier cosine integral transform defined by Z∞ γ 1 f ∗ g (x) = √ f (t)[g(x + α + t) 2 2π 0 + 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 γ ∀y ∈ R+ . Fc f ∗ g (y) = cos αy Fc f (y). Fc g (y), (2.2) Further, convolution operator (2.1) is continuous, bounded and satisfies Par- seval type equality r Z∞ γ 2 f ∗ g (x) = Fc f (y). Fc g (y) cos αy cos xy dy. (2.3) π 0 Definition 2.2. The functional spaces Lp (R+ ) and Lpα,β R+ are respestively defined by n Z∞ p1 o p p L (R+ ) = f : | f (x) | dx < +∞ ; (2.4) 0 and n Z∞ 1p o Lpα,β R+ = f: γ −βx t e p | f (x) | dx < +∞ . (2.5) 0 92
The convolution with a weight function related to the Fourier cosine integral transform 1 1 1 Theorem 2.2. (A Young’s type theorem) Let p, q, r > 1, such that + + = 2, p q r and let f ∈ Lp (R+ ), g ∈ Lq (R+ ), h ∈ Lr (R+ ), then  Z∞  r  γ  2  f ∗ g (x) . h(x) dx ≤  kf k Lp (R+ ) kgk Lq (R+ ) khk Lr (R+ ) . (2.6)  π 0 Proof. In view of the convolution of two functions f (x) and g(x) with a weight function defined in (1.7), we have  Z∞   γ    f ∗ g (x) . h(x) dx  0 ∞ ∞ 1 Z Z ≤ √ |f (t)| |g(x + α + t) + g(|x + α − t|)| |h(x)| dt dx 2 2π 0 0 Z∞ Z∞ 1 + √ |f (t)| |g(|x − α + t|) + g(|x − α − t|)| |h(x)| dt dx. (2.7) 2 2π 0 0 Let p1 , q1 , r1 be the conjugate exponentials of p, q, r respestively, it means 1 1 1 1 1 1 + = 1, + = 1, + = 1. p p1 q q1 r r1 It is obvious that 1 1 1 + + = 1. p1 q1 r1 Put q r U(x, y) = |g(x + α + t) + g(|x + α − t|)| p1 . |h(x)| p1 ; p r V (x, y) = |f (t)| q1 . |h(x)| q1 ; p q W (x, y) = |f (t)| r1 . |g(x + α + t) + g(|x + α − t|)| r1 . 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 Z ∞Z ∞ |f (t)|. |g(x + α + t) + g(|x + α − t|)| |h(x)| dt dx 0 0 Z∞ Z∞ = U(x, y) . V (x, y) . W (x, y)dtdx. 0 0 93
Nguyen Xuan Thao and Ta Duy Cong Applying the H¨older’s inequality for three functions U(x, y) , V (x, y) , W (x, y), we have Z∞ Z∞ U(x, y) . V (x, y) . W (x, y)dydx ≤ kUkLp1 (R2 ) kV kLq1 (R2 ) kW kLr1 (R2 ) . + + + 0 0 It follows that Z ∞Z ∞ |f (t)| |g(x + α + t) + g(|x + α − t|)| |h(x)| dt dx 0 0 ≤ kUkLp1 (R2 ) kV kLq1 (R2 ) kW kLr1 (R2 ) . (2.9) + + + In the space Lp1 (R2+ ), we have Z∞ Z∞ kUkpL1p1 (R2 ) = |g(x + α + t) + g(|x + α − t|)|q . |h(x)|r dx dt + 0 0 Z∞ Z∞ = |g(x + α + t) + g(|x + α − t|)|q dt |h(x)|r dx. (2.10) 0 0 Note that |t|q is a convex function, hence Z∞ |g(x + α + t) + g(| x + α − t |)|q dt 0 Z∞ Z∞ Z∞ q−1 q q q ≤ 2 |g(x + α + t)| dt + |g(| x + α − t |)| dt = 2 | g(t) |q dt. (2.11) 0 0 0 From (2.11) and (2.10), we get Z∞ Z∞ w wp1 wU w p1 L (R2 ≤ 2 q q | g(t) | dt |h(x)|r dx = 2q kgkqLq (R+ ) khkrLr (R+ ) . (2.12) +) 0 0 Similarly, in the space Lr1 (R2+ ), we have w wr1 wW w r 2 ≤ 2q kf kp p q L 1 (R ) L (R+ ) kgkLq (R+ ) . (2.13) + In the space Lq1 (R2+ ), we have w w p r wV w q 2 = kf k q1p khk q1 L 1 (R ) L (R+ ) 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 have w w w w w w wU w p 2 wV w q 2 wW w r 2 ≤ 2. kf kLp (R ) kgkLq (R ) khkLr (R ) . (2.15) L (R ) 1 +L (R ) 1 L 1 (R ) + + + + + From (2.15) and (2.9), we have
Z∞ Z∞
f (t)[g(x + α + t) + g(| x + α − t |)] h(x) dt dx
0 0 ≤ 2. kf kLp (R+ ) kgkLq (R+ ) khkLr (R+ ) . (2.16) Similarly Z∞ Z∞ |f (t)| |g(|x − α + t|) + g(|x − α − t|)| h(x) dt dx 0 0 ≤ 2. kf kLp (R+ ) kgkLq (R+ ) khkLr (R+ ) . (2.17) From formulas (2.16), (2.17) and (2.7), we get  Z∞  r  γ  2  f ∗ g (x) . h(x) dx  ≤ kf kLp (R+ ) kgkLq (R+ ) khkLr (R+ ) .  π 0 The proof is complete. 1 1 1 Corollary 2.1. Let p, q, r > 1 such that + = 1 + , and let f ∈ Lp (R+ ), g ∈ p q r γ q r L (R+ ), then f ∗ g (x) ∈ L (R+ ) and r γ 2 k f ∗ g kLr (R+ ) ≤ kf kLp (R+ ) kgkLq (R+ ) . (2.18) π Theorem 2.3. Let f ∈ Lp (R+ ), g ∈ Lq (R+ ), 0 < β ≤ 1, p, q > 1 such that 1 1 + = 1. Then the convolution ( 2.1) exists, is continuous and bounded. Further- p q γ more, f ∗ g ∈ Lrα,γ (R+ ) and the following estimation holds γ 2q k(f ∗ g)kLrα,γ (R+ ) ≤ √ C kf kLp (R+ ) kgkLq(R+ ) , (2.19) 2π γ+1 1 where C = β − r .Γ 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 H¨older’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 Z∞ λ(x) f (x) + √ f (t) ψ(g)(x, t) dt = h(x). (3.1) 2 2π 0 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 1 ψ(g)(x, t) = g(x + α + t) + g(|x + α − t|) + g(|x + α − t|) + g(|x − α − t|) . (3.2) λ(t) Theorem 3.1. Suppose that 1 + cos αy Fc g (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 h γ f (x) = h(x) − ∗ ϕ (x).λ(x). (3.3) λ where ϕ(x) is a continuous function, belongs to L(R+ ) defined by Fc g (y) Fc ϕ (y) = . 1 + cos αy Fc g (y) Proof. We can rewrite equation (3.1) as follows f γ f (x) + λ(x) ∗ g (x) = h(x). (3.4) λ The given condition shows that λ(x) 6= 0 for all x ∈ R+ , hence the equation (3.4) is equivalent to f (x) f γ h(x) + ∗ g (x) = . (3.5) λ(x) λ λ(x) Applying the Fourier cosine integral transform to both sides of (3.5) and using the factorization equality (2.2), we get f h i h Fc (y) 1 + cos αy Fc g (y) = Fc (y). λ λ By virtue of the hypothesis, we have h f F c (y) Fc (y) = λ , λ 1 + cos αy Fc g (y) 96
The convolution with a weight function related to the Fourier cosine integral transform or equivalently, f h h cos αy Fc g (y) i Fc (y) = Fc (y) 1 − . (3.6) λ λ 1 + cos αy Fc g (y) Due to Wiener-Levy’s theorem [1], there exists a unique function ϕ(x) ∈ L(R+ ) such that Fc g (y) Fc ϕ (y) = . 1 + cos y Fc g (y) Therefore equation (3.6) becomes f h h i Fc (y) = Fc (y) 1 − cos αy Fc ϕ (y) λ λ h h γ = Fc (y) − Fc ∗ ϕ (y) (3.7) λ λ Applying the inverse Fourier cosine integral transform to both sides of equation (3.7), we get f (x) h(x) h γ = − ∗ ϕ (x). λ(x) λ(x) λ It follows the solution of equation (3.1) h γ f (x) = h(x) − ∗ ϕ (x).λ(x). λ h Note that conditions m < |λ(x)| and h ∈ L(R+ ) imply that ∈ L(R+ ). By λ 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  Z∞   λ 1 (x) f (x) + √ g(t) h(ϕ)(x, t)dt = p (x)      2 2π 0 Z∞ (3.8)   λ2 (x)  √  f (t) k(ψ)(x, t)dt + g(x) = q(x),  2 2π   0 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 1 h(ϕ)(x, t) = ϕ(x+α+t)+ϕ(| x+α−t |)+ϕ(| x−α+t |)+ϕ(| x−α−t |) , (3.9) λ2 (t) 97
Nguyen Xuan Thao and Ta Duy Cong 1 k(ψ)(x, t) = ψ(x + α + t) + ψ(| x + α − t |) + ψ(| x − α + t |) + ψ(| x − α − t |) . λ1 (t) (3.10) γ Theorem 3.2. Suppose that the condition 1 − cos αy Fc ϕ ∗ 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 q γ h p q γ γ i f (x) = p(x) − λ1 (x). ∗ ϕ (x) + λ1 (x). − ∗ ϕ ∗ ξ (x), λ2 λ1 λ2 p γ h q p γ γ i (3.11) g(x) = q(x) − λ2 (x). ∗ ψ (x) + λ2 (x). − ∗ ψ ∗ ξ (x), λ1 λ2 λ1 Here, ξ(x) is a unique continuous function belonging to L(R+ ) and defined by γ Fc ϕ ∗ ψ (y) Fc ξ (y) = γ . (3.12) 1 − cos αy Fc ϕ ∗ ψ (y) Proof. The equation system (3.8) can be rewritten as follows  g γ f (x) + λ1 (x).  ∗ ϕ (x) = p (x) λ2 f γ (3.13) λ2 (x).  ∗ ψ (x) + g(x) = q(x). λ1 Applying Fourier cosine transform Fc to both sides of each equations of (3.13), we obtain f g p  Fc  (y) + cos αy Fc (y) Fc ϕ (y) = Fc (y) λ1 λ2 λ1 (3.14) f g q cos αy Fc  (y) Fc ψ (y) + Fc (y) = Fc (y). λ1 λ2 λ2 Under the condition of the hypothesis, it follows Cramer linear equation system (3.14) having a unique solution  p q γ F c − ∗ ϕ (y) f  λ1 λ2   Fc λ1 (y) =  γ   1 −cos αy Fc ϕ ∗ ψ (y) q p γ  Fc − ∗ ψ (y) g λ2 λ1   Fc λ (y) = ,    γ 2 1 − cos αy Fc ϕ ∗ ψ (y) or equivalently,  γ  f p q γ h cos αy Fc ϕ ∗ ψ (y) i Fc λ1 (y) = Fc λ1 − λ2 ∗ ϕ (y). 1 +  γ   1 − cos αy Fc ϕ ∗ ψ (y) γ  g q p γ h cos αy F c ϕ ∗ ψ (y) i F (y) = F − ∗ ψ (y). 1 + .    c c γ λ2 λ2 λ1  1 − cos αy Fc ϕ ∗ 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 γ Fc ϕ ∗ ψ (y) Fc ξ (y) = γ . 1 − cos αy Fc ϕ ∗ ψ (y) Therefore f hp q γ i  Fc  (y) = Fc − ∗ ϕ (y). 1 + cos αy Fc ξ (y) λ1 h λq1 λ2 g p γ i Fc  (y) = Fc − ∗ ψ (y). 1 + cos αy Fc ξ (y) . λ2 λ2 λ1 We obtain f hp q γ i h p q γ γ i  Fc  (y) = Fc − ∗ ϕ (y) + Fc − ∗ ϕ ∗ ξ (y) λ1 λ1 λ2 λ 1 λ 2 g h q p γ i h q p γ γ i Fc  (y) = Fc − ∗ ψ (y) + Fc − ∗ ψ ∗ ξ (y). λ2 λ2 λ1 λ2 λ1 It shows that  f (x) p q γ h p q γ γ i = − ∗ ϕ (x) + − ∗ ϕ ∗ ξ (x)   λ1 (x) λ1 λ2 λ1 λ2  g(x) q p γ h q p γ γ i = − ∗ ψ (x) + − ∗ ψ ∗ ξ (x).   λ2 (x) λ2 λ1 λ2 λ1  Therefore, we obtain a unique solution of system (3.8)  q γ h p q γ γ i f (x) = p(x) − λ1 (x). ∗ ϕ (x) + λ1 (x). − ∗ ϕ ∗ ξ (x) λ2 λ1 λ2 g(x) = q(x) − λ2 (x). p ∗ ψ (x) + λ2 (x). q − p ∗ ψ ∗γ ξ (x) γ γ h i λ1 λ2 λ1 Conditions 0 < m ≤ |λj (x)|, j = 1, 2 and p, q ∈ L(R+ ) imply that λp1 , λq2 ∈ 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 cos αx, 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. Academic Press/Elsevier Science, New York/Amsterdam. 99
Nguyen Xuan Thao and Ta Duy Cong [3] Baterman H. and Erdelyi A., 1954. Tables of Intergral Transforms Vol. 1. Mc- Graw - Hill, New York, Toronto, London. [4] Debnath L. and Debnath D., 2007. Intergral Transforms and Applications. Chap- man Hall CRC, London Boca Raton. [5] Ditkin V. A. and Prudnikov A., 1974. Intergral Transformations and Operator Canculus. Moscow. [6] Erdelyi A. and Baterman H., 1954. Tables of Intergral Transforms, Vol. 1. Mc- Graw - Hill, New York, Toronto, London. [7] Hirchman I. I. and Widder O. V., 1955. The convolution Transform. Princeton, New Jersey. [8] Nguyen Thanh Hong, 2010. Inequalities for Fourier cosine convolution and ap- plications. Intergral Transforms and Special Funtions, Vol. 21, No. 10, pp. 755 - 763. [9] Kakichev V. A. and Nguyen Xuan Thao and Nguyen Thanh Hai, 1996. Com- position method to constructing convolutions for intergral transforms. Intergral Transforms and Special Funtions, No. 3, pp. 235-242. [10] Ruzuk I. M. and Gradstein I. S., 1951. Tables of Integrals, sum, series and products. I*L. Moscow. [11] Saitoh S., 1993. Inequalities in the most simple Sobolev space and Convolution of L2 Functions with weight. Proc. Amer. Math. Soc 118, pp 515-520. [12] Saitoh S., 2000. Weight L2 -norm Inequalities in Convolution, Survey on clas- sical Inequalities. Kluwer Academic Publishers, Amsterdam. [13] Sneddon I. E., 1951. Fourier Transforms. McGraw - Hill, New York. [14] Nguyen Xuan Thao and Nguyen Thanh Hai, 1997. Convolution for Intergral Transforms and Their Aplications. Moscow. [15] Nguyen Xuan Thao and Nguyen Minh Khoa, 2004. On the convolution with the weight funtion for the Fourier cosine Integral Transform. Acta Mathematica Vietnamica, Volume 29, Number 2, pp 149-162. [16] Titchmarch E. C., 1937. Introduction to Theory of Fourier Intergrals. Oxford Univ. Press. 100