86 On the generalized convolutions for Fourier cosine and sine transforms the convolution has the form [8] (f g)(x) = p 1 f(y)[g(j x ; y j)+g(x + y)]d

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1 East-West Journal of Mathematics: Vol. 1, No 1 (1998) pp ON THE GENERALIZED CONVOLUTIONS FOR FOURIER COSINE AND SINE TRANSFORMS Nguyen Xuan Thao, V.A. Kakichev Novgorod University, St. Petersburg str. 41 Novgorod 1733, Russia Vu Kim Tuan Department of Mathematics and Computer Science Faculty of Science, Kuwait University P.O. Box 5969, Safat 136, Kuwait Abstract A generalized convolution for the Fourier cosine and sine transforms is introduced, its properties and applications to integral equations are considered. 1. Introduction Generalized convolution of functions f and g under three operators K, K1, K, and with some weight-function is a function, denoted by the symbol f g, such that the following factorization property holds [5] : K(f g)(x) =(x)(k1f)(x)(kg)(x): (1:1) If K = K1 = K we have the usual classical convolution [3,4]. For example, for K = K1 = K = F c { the Fourier cosine transform [8] (F c f)(x) = r f(y)cos xy dy (1:) Key phrases. Fourier cosine and Fourier sine transforms, convolution, integral equations. 85

2 86 On the generalized convolutions for Fourier cosine and sine transforms the convolution has the form [8] (f g)(x) = p 1 f(y)[g(j x ; y j)+g(x + y)]dy (1:3) and the property (1.1) holds F c (f g)(x) =(F c f)(x)(f c g)(x): (1:4) Otherwise, there appear "exotic" convolutions. An example of generalized convolutions was rst introduced by Churchill [1] (f 1 g)(x) = p 1 f(y)[g(j x ; y j) ; g(x + y)]dy (1:5) and the respective factorization property (1.1) for (1.5) has the form F s (f 1 g)(x) =(F s f)(x)(f c g)(x) (1:6) where F s is the Fourier sine transform [8] r Z 1 (F s f)(x) = f(y)sin xy dy: (1:7) Many authors have studied similar convolutions for Hankel's transform [1], Stieltjes' transform [9], Hilbert's transform [], G-transform [7], and integral transforms of Mellin convolution type [6,1]. The present work is devoted to investigate properties of another generalized convolution for Fourier cosine and sine transforms, dierent from (1.5), and its application to a linear system of integral equations.. The generalized convolution Denition 1. A generalized convolution for the Fourier cosine and sine transforms is dened as follows: (f g)(x) = p 1 f(y)[sign(y ; x)g(j y ; x j)+g(y + x)]dy: (:1) Theorem 1. Let f g L(R+), then the convolution f g belongs to L(R+) and F c (f g)(x) =(F s f)(x)(f s g)(x) xr+ : (:)

3 N.X. Thao, V.A. Kakichev and V.K. Tuan 87 Proof. We have j (f g)(x) j dx p 1 j (f(y) j (j g(j x;y j) j + j g(x+y) j)dxdy p 1 j f(y) j dy j g(jxj) j dx + j g(x) j dx ;y y r Z 1 = j f(y) j dy j g(x) j dx < 1: Hence, the convolution (.) belongs to L(R+). Furthermore, (F s f)(x)(f s g)(x) = cos x(u;v) f(u)g(v)du dv; 1 sin xu sin xv f(u) g(v) du dv cos x(u+v) f(u)g(v)du dv cos xt [f(y)g(y + t)+f(y + t)g(y)]dy dt Z t ; 1 cos xt f(y)g(t ; y)dy dt Z t cos xt f(y)g(y + t)dy + f(y)g(y ; t)dy ; f(y)g(t ; y)dy dt t cos xt f(y)g(y + t)dy + sign(y ; t)f(y)g(jy ; tj)dy dt Theorem 1 is thus proved. = F c (f g)(x): Remark 1: Formulas (1.4) and (.) show thattheconvolution f g and f g are commutative. On the other hand, the convolution f 1 g is non commutative: f 1 g = ;g 1 r f + f g (:3) L where f g is the Laplace convolution L (f Z x L g)(x) = f(y) g(x ; y) dy: (:4)

4 88 On the generalized convolutions for Fourier cosine and sine transforms Indeed, we have (f 1 g)(x) = p 1 f(y)[g(j x ; y j) ; g(x + y)]dy = p 1 f(x + s)g(j s j) ds ; f(s ; x)g(s) ds ;x = p 1 g(s)[f(x + s) ; f(j s ; x j)] ds+ Z Z x + f(x + s) g(j s j) ds + f(j s ; x j) g(s) ds ;x = ;(g 1 r f)(x)+ (f L g)(x): Remark : Convolution (.1) was introduced implicitly, but incorrectly in [8], where the term sign(y ; x) was missing. Theorem. Let the functions f g h belong to L(R+). Then the following formulas hold x (f 1 g) 1 h =(f 1 h) 1 g = f 1 (g h) (:5) f (g h) =g (h 1 f) =h (g 1 f) (:6) f 1 (g h) =g 1 (f h) =h 1 (f g) (:7) f (g h) =g (f h) =h (f g) (:8) The proof follows easily from formulas (1.4), (1.6) and (.). For example, we have F s [(f 1 g) 1 h] =F s (f 1 g)f c (h) =F s (f)f c (g)f c (h) =[F s (f)f c (h)]f c (g) =F s (f 1 h)f c (g) = F s [(f 1 h) 1 g]: Hence, (f 1 g) 1 h =(f 1 h) 1 g. On the other hand, F s [(f 1 g) 1 h] =F s (f)[f c (g)f c (h)] = F s (f)f c (g h) =F s [(f 1 (g h)]: Therefore, (f 1 g) 1 h = f 1 (g h), and formula (.5) is proved. By the same way, one can verify the other parts, too.

5 N.X. Thao, V.A. Kakichev and V.K. Tuan Applications to integral equations We consider the following linear system of integral equations: '(x)+1 k(x y) (y)dy = f(x) (3:1) (x)+ h(x y)'(y)dy = g(x) x R+ : (3:) where ' and are unknown functions, f and g are given functions, 1 and denote complex parameters, and k(x y) andh(x y) arethekernels that can be expressed in the form k(x y) =k1(j x ; y j) ; k1(x + y) h(x y) = sign(y ; x)h1(j x ; y j)+h1(x + y): (3:3) Applying the Fourier sine transform to equation (3.1) and the Fourier cosine transform to equation (3.) and using the convolution formulas (1.6) and (.) we obtain a linear system of algebraic equations Suppose that F s (')+ p 1F c ( )F s (k1) =F s (f) F c ( )+ p F s (')F s (h1) =F c (g): (3:4) 1 ; 1(F s k1)(x)(f s h1)(x) 6= (3:5) for any x R+. Then the linear system (3.4) has the solution F s (') =[F s (f) ; p 1F c (g)f s (k1)]=[1 ; 1F s (k1)f s (h1)] F c ( )=[F c (g) ; p F s (f)f s (h1)]=[1 ; 1F s (k1)f s (h1)]: (3:6) Consider the function (t) =1t=(1 ; 1t) with t = F c (k1 h1) = F s (k1)f s (h1). Since (t) is analytic under the condition (3.5) and () =, by the Wiener-Levi theorem there exists a function l L(R+) such that Hence, we obtain F c (l) =1F s (k1)f s (h1)=[1 ; 1F s (k1)f s (h1)]: (3:7) F s (') =[F s (f) ; p 1F c (g)f s (k1)][1 + F c (l)] = F s (f) ; p 1F s (k1 1 g)+f s (f 1 l) ; p 1F s ((k1 1 g) 1 l):

6 9 On the generalized convolutions for Fourier cosine and sine transforms Therefore, ' = f ; p 1k1 1 g + f 1 l ; p 1(k1 1 g) 1 l: (3:8) Similarly, wehave F c ( )=[F c (g) ; p F s (f)f s (h1)][1 + F c (l)] = F c (g) ; p F c (f h1)+f c (l g) ; p F c (l (f h1)): Consequently, = g ; p f h1 + l g ; p l (f h1): (3:9) References [1] R.V. Churchill, "Fourier series and boundary value problems", New York,1941. [] H.J. Glaeske and Vu Kim Tuan, Convolution of the Hilbert transform and its application to some nonlinear singular integral equations, Integral Transforms and Special Functions, 3(1995), no. 4, [3] V.A. Kakichev, On the convolutions for integral transform (Russian), Vestsi Akad. Navuk BSSR, Ser. Fiz. Mat. Navuk, (1967), no., [4] V.A. Kakichev, On the matrix convolutions for power series (Russian). Izv. Vyssh. Uchebn. Zaved. Mat., (199), no., [5] V.A. Kakichev and Nguyen Xuan Thao, On the design method for the generalized integral convolutions (Russian), Izv. Vyssh. Uchebn. Zaved. Mat., (1997), no. 6. [6] Nguyen Thanh Hai and S.B. Yakubovich,The double Mellin-Barnes type integrals and their applications to convolution theory, World Scientic, Singapore, (199). [7] M. Saigo and S. B. Yakubovich, On the theory of convolution integrals for G-transforms, Fukuoka Univ. Sci. Reports, 1(1991), no., [8] I.N. Sneddon, "The use of integral transforms", McGray{Hill, New York, (197). [9] H.M. Srivastava and Vu Kim Tuan, A new convolution theorem for the Stieltjes transform and its application to a class of singular integral equations, Arch. Math., 64(1995), no., [1] Vu Kim Tuan and Megumi Saigo, Convolution of Hankel transform and its application to an integral involving Bessel function of rst kind, Internat. J. Math. Math. Sci., 18(1995), no. 3, [11] S.B. Yakubovich, On the constructive method of integral convolutions (Russian), Dokl. Akad. Nauk BSSR, 34(199), no. 7, [1] S.B. Yakubovich and Yu.F. Luchko, "The hypergeometric approach to integral transforms and convolutions", Kluwer Acad. Publ., 1994.

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

Mathematical Problems in Engineering Volume, Article ID 7967, 6 pages doi:.55//7967 Research Article On the Polyconvolution with the Weight Function for the Fourier Cosine, Fourier Sine, and the Kontorovich-Lebedev