A generalized convolution with a weight function for the Fourier cosine
and sine transforms is introduced. Its properties and applications to solving
a system of integral equations are considered

Kim Tuan, Vu

Minh Khoa, Nguyen

Xuan Thao, Nguyen

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

A GENERALIZED CONVOLUTION WITHA WEIGHT FUNCTION FOR THE FOURIERCOSINE AND SINE TRANSFORMSNguyen Xuan Thao 1, Vu Kim Tuan 2, Nguyen Minh Khoa 3Dedicated to Professor Ivan H. Dimovskion the occasion of his 70th birthdayAbstractA generalized convolution with a weight function for the Fourier cosineand sine transforms is introduced. Its properties and applications to solvinga system of integral equations are considered.2000 Mathematics Subject Classification: 42A38, 42A76, 42A85, 42A96Key Words and Phrases: convolution, Fourier cosine transform, Fouriersine transform, integral equation1. IntroductionLet Fc and Fs be the Fourier cosine and sine transforms, respectively:Fc(f)(x) =1√2pi∞∫0f(y) cosxy dy, x > 0, (1)Fs(f)(x) =1√2pi∞∫0f(y) sinxy dy, x > 0. (2)324 Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh KhoaA convolution of two functions f and g for the Fourier cosine transform wasgiven in 1941 by Churchill [1]:(f ∗Fcg)(x) =1√2pi∞∫0f(y) (g(|x− y|) + g(x+ y)) dy, x > 0 (3)with the factorization propertyFc(f ∗Fcg)(y) = (Fcf)(y) (Fcg)(y), ∀y > 0. (4)A convolution with the weight function γ(y) = sin y of functions f and gfor the Fourier sine transform Fs was studied in [2], [7]:(f ∗Fsg)(x) =12√2pi∞∫0f(t) (g(x+ t+ 1) + g(|x− t+ 1|)sign(x− t+ 1)(5)+ g(|x+ t− 1|)sign(x+ t− 1) + g(|x− t− 1|)sign(x− t− 1)) dt, x > 0,for which the factorization property holds:Fs(f ∗Fsg)(y) = sin y (Fsf)(y) (Fsg)(y), ∀y > 0. (6)A generalized convolution for the Fourier sine and cosine transforms wasfirst introduced by Churchill in [1]:(f ∗1g)(x) =1√2pi∞∫0f(y) (g(|x− y|)− g(x+ y)) dy, x > 0, (7)that satisfies the factorization propertyFs(f ∗1g)(y) = (Fsf)(y)(Fcg)(y), y > 0. (8)Yakubovich et al. [11], [12], [13] studied some special cases of generalizedconvolutions for nonconvolution integral transforms. In 1998, Kakichev andNguyen Xuan Thao [3] proposed a constructive method of defining a general-ized convolution for three (possibly different) integral transformsK1,K2,K3with a weight function γ(x) with a factorization propertyK1(fγ∗g)(x) = γ(x)(K2f)(x)(K3g)(x).In recent years, generalized convolutions, for instance, for the transformsof Stieltjes [15], Hilbert [6], [14], Hankel [16], H-transforms [4], I-transformsA GENERALIZED CONVOLUTION WITH . . . 325[9], the Fourier cosine and sine transforms [8] have been studied. For ex-ample, a generalized convolution for the Fourier cosine and sine transformshas been introduced in [8], as follows:(f ∗2g(x) =1√2pi∞∫0f(y) (sign(y − x)g(|y − x|) + g(y + x)) dy, x > 0. (9)It satisfies a factorization propertyFc(f ∗2g)(y) = (Fsf)(y)(Fsg)(y), ∀y > 0. (10)In this paper we give a new generalized convolution with a weight func-tion for the Fourier cosine and sine transforms. We prove some of its prop-erties as well as we point out some of its relationships to several well-knownconvolutions and apply this notion to solve a system of integral equations.2. A generalized convolutionDefinition 1. A generalized convolution with the weight functionγ(y) = sin y for the Fourier cosine and sine transforms of functions f and gis defined by(fγ∗g)(x) = 12√2pi∞∫0f(y) [(g(|x+ y − 1|) + g(|x− y + 1|)− g(x+ y + 1)− g(|x− y − 1|)) dy, ∀x > 0. (11)We denote by L(R+) the set of all functions f defined on (0,∞) such that∞∫0|f(x)|dx <∞.Theorem 1. Let f, g ∈ L(R+). Then the generalized convolution withthe weight-functions γ(y) = sin y for the Fourier cosine and sine transformsbelongs to L(R+), and the following factorization property holds:Fc(fγ∗g)(y) = sin y (Fsf)(y)(Fcg)(y), ∀y > 0. (12)P r o o f. Based on (11) and the fact that f and g ∈ L(R+) we have∞∫0|(f γ∗g)|dx ≤ 12√2pi∞∫0∞∫0|f(y)| . |[g(|x+ y − 1|) + g(|x− y + 1|)326 Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh Khoa− g(x+ y + 1)− g(|x− y − 1|)]| dydx ≤ 12√2pi∞∫0|f(y)|[ ∞∫0|g(|x+y−1|)|dx+∞∫0|g(|x− y − 1|)|dx +∞∫0|g(x+ y + 1)| dx++∞∫0|g(x− y − 1)| dx dy.(13)On the other hand,∞∫0|(g(x+ y + 1)| dx+∞∫0|g(|x− y − 1|)| dx=∞∫y+1|g(t)| dt+∞∫−y−1|g(|t|)| dt = 2∞∫0|g(t)| dt. (14)Similarly,∞∫0|g(|x− y + 1|)| dx+∞∫0|g(|x+ y − 1|)| dx=∞∫1−y|g(|t|)| dt+∞∫y−1|g(|t|)| dt = 2∞∫0|g(t)| dt. (15)¿From (13), (14) and (15) it follows that∞∫0∣∣∣(f γ∗g)(x)∣∣∣ dx ≤√ 2pi∞∫0|f(t)|dt∞∫0|g(t)|dt <∞.So(fγ∗g)(x) ∈ L(R+).Now we prove the factorization property (12). Sincesinx (Fsf) (x) (Fcg) (x) =2pi∞∫0∞∫0sinx sinxu cosxv f(u) g(v) dudvandsinx sinxu cosxv =14[cosx(u− v − 1) + cosx(u+ v − 1)A GENERALIZED CONVOLUTION WITH . . . 327− cosx(u+ v + 1)− cosx(u− v + 1)] ,we obtainsinx (Fsf) (x) (Fcg) (x) =12pi∞∫0∞∫0[cosx(u− v − 1) + cosx(u+ v − 1)− cosx(u+ v + 1)− cosx(u− v + 1)] f(u)g(v)dudv. (16)Taking a substitution y = u, t = u+ v + 1, we get12pi∞∫0∞∫0cosx(u+v+1)f(u)g(v)dudv =12pi∞∫0∞∫y+1cosxt f(y)g(t−y−1)dtdy=12pi∞∫0∞∫0cosxt f(y)g(|t−y−1|)dtdy− 12pi∞∫0y+1∫0cosxt f(y)g(y−t+1)dtdy.(17)Similarly, with a substitution y = u, t = v − u− 1, we have12pi∞∫0∞∫0cosx(u−v+1) f(u)g(v)dudv = 12pi∞∫0∞∫−y−1cosxt f(y)g(t+y+1)dtdy=12pi∞∫0∞∫0cosxt f(y)g(t+y+1)dtdy+12pi∞∫00∫−y−1cosxt f(y)g(t+y+1)dtdy.(18)Furthermore,∞∫00∫−y−1cosxt f(y)g(t+ y + 1)dtdy = −∞∫0∫ 0y+1cosxt f(y)g(y − t+ 1)dtdy=∫ ∞0∫ y+10cosxt f(y)g(y − t+ 1)dtdy. (19)¿From (17), (18) and (19) we obtain− 12pi∫ ∞0∫ ∞0[cosx(u+ v + 1) + cosx(u− v + 1)] f(u)g(v)dudv328 Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh Khoa= − 12pi∫ ∞0∫ ∞0cosxt [g(|t− y − 1|) + g(t+ y + 1)] f(y)dtdy. (20)Similarly,12pi∫ ∞0∫ ∞0cosx(u+ v − 1)f(u)g(v)dudv = 12pi∫ ∞0∫ ∞y−1cosxtf(y)× g(t− y + 1)dtdy = 12pi∫ ∞0∫ ∞0cosxtf(y)× g(|t− y + 1|)dtdy − 12pi∫ ∞0∫ y−10cosxt f(y) g(|t− y + 1|)dtdy. (21)With a substitution y = u, t = v − u+ 1 we get12pi∫ ∞0∫ ∞0cosx(u− v − 1)f(u)g(v)dudv = 12pi∫ ∞0∫ ∞1−ycosxt f(y)× g(t+ y − 1)dtdy = 12pi∫ ∞0∫ ∞0cosxt f(y)× g(|t+ y − 1|)dtdy − 12pi∫ ∞0∫ 1−y0cosxt f(y) g(|t+ y − 1|)dtdy. (22)On the other hand,∫ ∞0∫ y−10cosxtf(y)g(|t−y+1|)dtdy=−∫ ∞0∫ 01−ycosxtf(y)g(|t+y−1|)dtdy.(23)¿From (21), (22), and (23) we have12pi∫ ∞0∫ ∞0[cosx(u+ v − 1) + cosx(u− v − 1)] f(u)g(v)dudv=12pi∫ ∞0∫ ∞0cosxt [g(|t− y + 1|) + g(|t+ y − 1|)] f(y)dtdy. (24)Finally, from (16), (20) and (24),sinx(Fsf)(x)(Fcf)(x) =12pi∞∫0cosxy{∫ ∞0f(y) (25)× [g(|x+ y − 1|) + g(|x− y + 1|)− g(x+ y + 1)− g(|x− y − 1|)] dy}dx.The last equality and (11) yieldsinx (Fsf)(x)(Fcg)(x) = Fc(fγ∗g)(x).The proof is complete.A GENERALIZED CONVOLUTION WITH . . . 329Theorem 2. In the space L(R+) the generalized convolution (11) isnoncommutative:(fγ∗g)(x) = −(g γ∗f)(x)+ 12pi[sign(1−x)(f ∗Lg)(|x−1|)+(f ∗Lg)(x+1)]. (26)Here f ∗Lg denotes the convolution for the Laplace integral transform(f ∗Lg)(x) =∫ x0f(t) g(x− t) dt.P r o o f. Indeed, with the substitutions t = x+y−1, t = y−x−1, t =x+ y + 1, t = y − x+ 1, respectively, we have(fγ∗g)(x) = 12√2pi[ ∫ ∞x−1g(|t|)f(t− x+ 1)dt+∫ ∞−(x+1)g(|t|)f(t+ x+ 1)dt−∫ ∞x+1g(|t|)f(t− x− 1)dt−∫ ∞1−xg(|t|)f(t+ x− 1)dt](27)=12√2pi{∫ ∞0[f(|t− x+ 1|)+f(t+ x+ 1)−f(|t− x− 1|)−f(|t+ x− 1|)]× g(t)dt+∫ 0x−1g(|t|)f(t− x+ 1)dt+∫ 0−(x+1)g(|t|)f(t+ x+ 1)dt−∫ 0x+1g(|t|)f(t− x− 1)dt−∫ 01−xg(|t|)f(t+ x− 1)dt}=12√2pi{∫ ∞0[f(|x− t− 1|) + f(x+ t+ 1)− f(|x− t+ 1|)− f(|x+ t− 1|)]× g(t)dt+∫ 0x−1g(|t|)f(t− x+ 1)dt+∫ 0−(x+1)g(|t|)f(t+ x+ 1)dt−∫ 0x+1g(|t|)f(t− x− 1)dt−∫ 01−xg(|t|)f(t+ x− 1)dt}= −(g γ∗f)(x) + 12√2pi{∫ 0x−1g(|t|)f(t− x+ 1)dt−∫ 01−xg(|t|)f(t+x−1)dt−∫ 0x+1g(|t|)f(t−x−1)dt+∫ 0−(x+1)g(|t|)f(t+x+1)dt}.With the substitution u = −t we get330 Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh Khoa∫ x−10g(|t|)f(|t− x+ 1|)du = −∫ 1−x0g(|u|)f(|u+ x− 1|)dv∫ −(x+1)0g(|t|)f(t+ x+ 1)du = −∫ 1+x0g(u)f(|u− x− 1|)du,therefore,∫ 1−x0g(|u|)f(|u+ x− 1|)du−∫ x−10g(|u|)f(|u− x+ 1|)dv=2∫ 1−x0 g(u)f(1− x− u)du, x < 1−2x−1∫0g(u)f(x− 1− u)du, x ≥ 1= 2 sign(1− x)|x−1|∫0g(u)f(|x− 1| − u)du = 2 sign(1− x)(f ∗Lg)(|x− 1|).Hence and with the substitutions u = t−x+1, u = t+x− 1, u = t−x− 1,u = t+ x+ 1 we obtain∫ 0x−1g(|t|)f(t− x+1)dt−∫ 01−xg(|t|)f(t+ x− 1)dt−∫ 0x+1g(t)f(t− x− 1)dt+∫ 0−(x+1)g(|t|)f(t+ x+ 1)dt =∫ 1−x0f(|u|)g(|u+ x− 1|)du−∫ x−10f(|u|)g(|u− x+ 1|)du−∫ −(x+1)0f(|u|)g(|u+ x+ 1|)du+∫ x+10f(u)g(|u− x− 1|)du = 2 sign(1− x)(f ∗Lg)(|x− 1|)+ 2(x+1)∫0f(u)g(x+ 1− u)du = 2 sign(1− x)(f ∗Lg)(|x− 1|) + 2(f ∗Lg)(x+ 1).Therefore,(fγ∗g)(x) = −(g γ∗f)(x) + 1√2pi[sign(1− x)(f ∗Lg)(|x− 1|) + (f ∗Lg)(x+ 1)].The theorem is proved.A GENERALIZED CONVOLUTION WITH . . . 331Theorem 3. In the space L(R+) the generalized convolution (11) isnot associative and satisfies the following equalities:a) fγ∗(g γ∗h) = g γ∗(f γ∗h);b) fγ∗(g γ∗h) = (f ∗Fsg)γ∗h, where (f ∗Fsg) is the Fourier-sine convolutionwith a weight-function (5);c) fγ∗(g ∗Fsh) = (fγ∗g) ∗Fch, where (g ∗Fch) is the convolution for the Fouriercosine transform (3);d) fγ∗(g ∗2h) = (f ∗Fsg) ∗2h, where (g ∗2h) is the generalized convolution witha weight-function for the Fourier cosine and sine transforms (9);e) f ∗1(gγ∗h) = (f ∗1h)γ∗g, where (f ∗1h) is the generalized convolution for theFourier sine and cosine transforms (7).P r o o f. a) From the factorization property, we haveFc(fγ∗(g γ∗h))(x) = sinx(Fsf)(x)Fc(gγ∗h)(x)= sinx (Fsf)(x) sinx(Fsg)(x) (Fch)(x) = sinx (Fsg)(x)Fc(fγ∗h)(x)= Fc{gγ∗(f γ∗h)}(x), ∀x > 0,which implies that(fγ∗g) γ∗h = g γ∗(f γ∗h).b)Fc(fγ∗(g γ∗h))(x) = sinx(Fsf)(x)Fc(gγ∗h)(x)= sinx(Fsf)(x) sinx(Fsg)(x) (Fch)(x) = Fs(f ∗Fsg)(x) sinx (Fch)(x)= Fc{(f ∗Fsg)(x)γ∗h}(x), ∀x > 0.Hence fγ∗(g γ∗h) = (f ∗Fsg)γ∗h.The proofs for c), d), e) are similar to those of a) and b). The theoremis proved.332 Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh KhoaTheorem 4. The generalized convolution (11) is related to the convo-lution (7) as follows(fγ∗g)(x) = 12[(f ∗1g)(x+ 1)− (f ∗1g)(x− 1) sign(x− 1)]. (28)P r o o f. Indeed,i) For x ≥ 1, from Definition 1 we obtain(fγ∗g)(x) = 12{ 1√2pi∫ ∞0[g(|x− y + 1|)− g(x+ y + 1)] f(y)dy−12{ 1√2pi∫ ∞0[(|x− y − 1|)− g(x+ y − 1)] f(y)dy}=12[(f ∗1g)(x+ 1)− (f ∗1g)(x− 1)].ii) Let 0 < x < 1. We have(fγ∗g)(x) = 12{ 1√2pi∫ ∞0[g(|x− y + 1|)− g(x+ y + 1)] f(y)dy− 1√2pi∫ ∞0[g(|x− y − 1|)− g(|x+ y − 1|)] f(y)dy}=12{ 1√2pi∞∫0[g(|x− y + 1|)− g(x+ y + 1)] f(y)dy− 1√2pi∫ ∞0[g(|x− y − 1|)− g(|x+ y − 1|)] f(y)dy}=12[(f ∗1g)(x+ 1) + (f ∗1g)(1− x)]=12[(f ∗1g)(x+ 1)− (f ∗1g)(|x− 1|)sign(x− 1)].Hence(fγ∗g)(x) = 12[(f ∗1g)(x+ 1)− (f ∗1g)(|x− 1|)sign(x− 1)].The theorem is proved.Theorem 5. In the space L(R+) the generalized convolution (11) doesnot have a unit element.A GENERALIZED CONVOLUTION WITH . . . 333P r o o f. Suppose that there exists a unit element of the generalized con-volution (11) in L(R+) : eγ∗g = g γ∗e = g, for any function g ∈ L(R+). It fol-lows that Fc(eγ∗g)(y) = (Fcg)(y), ∀y > 0. Hence, sin y (Fse)(y) (Fcg)(y) =(Fcg)(y), ∀y > 0. The last equation is equivalent to the equality(Fcg)(y)[sin y(Fse)(y)− 1] = 0, ∀y > 0for any function g(y) ∈ L(R+). Choosing g such that (Fcg)(y) 6= 0, ∀y > 0,we see that sin y(Fse)(y)− 1 = 0 or (Fse)(y) = 1sin y , ∀y > 0.Thus, (Fse)(y) does not approach 0 as y tends to infinity, as it must beif e ∈ L(R+), and so e 6∈ L(R+). This is a contradiction. The theorem isproved.Set L(R+, ex) = {h(x), for all exh(x) ∈ L(R+)}Theorem 6. (Titchmarch-type theorem) Let f, g ∈ L(R+, ex). If(fγ∗g)(x) = 0, ∀x > 0,then either f(x) = 0 or g(x) = 0, ∀x > 0.P r o o f. Under the hypothesis (fγ∗g)(x) = 0, ∀x > 0, it follows thatFc(fγ∗g)(y) = 0, ∀y > 0. By virtue of Theorem 1,sin y(Fsf)(y)(Fcg)(y) = 0, ∀y > 0. (29)Consider the Fourier cosine integral transform(Fcg)(y) =√2pi∫ ∞0g(x) cos(yx)dx, y ∈ R+.Since ∣∣∣ dndyn(cos(yx)g(x))∣∣∣ = ∣∣∣g(x)xn cos(yx+ npi2)∣∣∣ ≤ |g(x)xn|= |e−xxng1(x)| = |e−xxn| |g1(x)| ≤ C|g1(x)|, g1(x) = ex g(x) ∈ L(R+),for x large enough, due to Weierstrass’ criterion, the integral∫ ∞0dndyn[cos(yx)g(x)] dxuniformly converges on R+. Therefore, based on the differentiability ofintegrals depending on parameter, we conclude that (Fcg)(y) is analytic.Similarly, (Fsf)(y) is analytic. So from (29) we have (Fsf)(y) = 0, ∀y > 0or (Fcg)(y) = 0, ∀y > 0. It follows that either f(x) = 0 or g(x) = 0, ∀x >0. The theorem is proved.334 Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh Khoa3. A system of integral equationsConsider a system of integral equationsλ11f(y) + λ12∞∫0ϕ(t)g1(y, t)dt = k(y),λ21∞∫0f(t)ψ1(y, t)dt+ λ22g(y) = h(y), y > 0.(30)Here, λ11, λ12, λ21, λ22 are complex numbers and ϕ,ψ, k, h ∈ L(R+), f andg are unknown functions of L(R+) andg1(y, t) =1√2pi[g(|t− y|) + g(t+ y)] ,ψ1(y, t)=1√2pi[ψ(|y + t− 1|)+ψ(|y − t+ 1|)−ψ(y + t+ 1)−ψ(|y − t− 1|)] .Theorem 7. Let 1 + C sign y (Fsϕ)(y)(Fcψ)(y) 6= 0. Then there existsa solution in L(R+) of (30) which is defined byf(y)=1λ{λ22k(y)− λ12(ϕ ∗1h)(y)− (k ∗1l)(y) + λ12((ϕ ∗1h) ∗1l)(y)}∈ L(R+),g(y)=1λ{λ11h(y)−λ21(kγ∗ψ)(y)−λ11(h ∗Fcl)(y)+λ21((kγ∗ψ) ∗Fcl)(y)}∈ L(R+).Here, l ∈ L(R+) is defined by(Fcl)(y) =C Fc(ϕγ∗ψ)(y)1 + CFc(ϕγ∗ψ)(y), λ = λ11λ22 , C = −λ12λ21λ11λ22.P r o o f. System (30) can be written in the form{λ11f(y) + λ12(ϕ ∗1g)(y) = k(y),λ21(fγ∗ψ)(y) + λ22g(y) = h(y), y > 0.Using the factorization properties of the convolutions (7) and (11) we have{λ11(Fsf)(y) + λ12(Fsϕ)(y)(Fcg)(y) = (Fsk)(y),λ21 sin y(Fsf)(y)(Fcψ)(y) + λ22(Fcg)(y) = (Fch)(y), y > 0.Since,A GENERALIZED CONVOLUTION WITH . . . 335∆ =∣∣∣∣ λ11 λ12(Fsϕ)(y)λ21(Fcψ)(y) sin y λ22∣∣∣∣= λ11λ22 − λ12λ21 sin y(Fsϕ)(y)(Fcψ)(y)= λ11λ12[1− λ12λ21λ11λ22sin y(Fsϕ)(y)(Fcψ)(y)]= λ[1 + C sign y(Fsϕ)(y)(Fcψ)(y)(y)] 6= 0,∆1 =∣∣∣∣(Fsk)(y) λ12(Fsϕ)(y)(Fch)(y) λ22∣∣∣∣ = (Fsk)(y)λ22 − λ12(Fsϕ)(y)(Fch)(y).Therefore,(Fsf)(y) =∆1∆=∆1λ− ∆1λC sin y(Fsϕ)(y)(Fcψ)(y)1 + C sin y(Fsϕ)(y)(Fcψ)(y),=∆1λ− ∆1λCFc(ϕγ∗ψ)(y)1 + CFc(ϕγ∗ψ)(y), y > 0.Due to Wiener-Levi’s theorem [7] there exists a function l ∈ L(R+) suchthat(Fcl)(y) =CFc(ϕγ∗ψ)(y)1 + CFc(ϕγ∗ψ)(y).It follows that(Fsf)(y) =1λ[∆1 −∆1(Fcl)(y)] = 1λ{(Fsk)(y)λ22−λ12(Fsϕ)(y)(Fch)(y)− [(Fsk)(y)λ22 − λ12(Fsϕ)(y)(Fch)(y)] (Fcl)(y)}=1λ{(Fsk)(y)λ22 − λ12Fs(ϕ ∗1h)(y)− Fs(k ∗1l)(y) + λ12Fs(ϕ ∗1h)(y)(Fcl)(y)}=1λ{(Fsk)(y)λ22 − λ12Fs(ϕ ∗1h)(y)− Fs(k ∗1l)(y) + λ12Fs((ϕ ∗1h) ∗1l)(y)}.Hence,f(y) =1λ{k(y)λ22 − λ12(ϕ ∗1h)(y)− (k ∗1l)(y) + λ12((ϕ ∗1h) ∗1l)(y)}.Similarly,∆2 =∣∣∣∣ λ11 (Fsk)(y)λ21 sin y(Fcψ)(y) (Fch)(y)∣∣∣∣336 Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh Khoa= λ11(Fch)(y)−λ21 sin y (Fsk)(y)(Fcψ)(y)=λ11(Fch)(y)−λ21Fc(kγ∗ψ)(y).It follows that(Fcg)(y) =∆2∆=∆2λ− ∆2λ(Fcl)(y) =1λ{λ11(Fch)(y)−λ21Fc(kγ∗ψ)(y)− λ11(Fch)(y)(Fcl)(y) + λ21Fc(kγ∗ψ)(y)(Fcl)(y)}=1λ{λ11(Fch)(y)− λ21Fc(kγ∗ψ)(y)−λ11(Fc(h ∗Fcl))(y) + λ21Fc((kγ∗ψ) ∗Fcl)(y)}.Henceg(y) =1λ{λ11h(y)− λ21(kγ∗ψ)(y)− λ11(h ∗Fcl)(y) + λ21((kγ∗ψ) ∗Fcl)(y)}.The proof is complete.References[1] R. V. Churchill, Fourier Series and Boundary Value Problems. NewYork (1941).[2] V. A. Kakichev, On the convolution for integral transforms (in Rus-sian). Izv. AN BSSR, Ser. Fiz. Mat., No 2 (1967), 48-57.[3] V. A. Kakichev and Nguyen Xuan Thao, On a constructive methodfor the generalized integral convolution (in Russian). Izv. Vuzov. Mat.,No 1(1998), 31-40.[4] V. A. Kakichev and Nguyen Xuan Thao, On the generalized convo-lution for H- transforms (in Russian). Izv. Vuzov Mat. No 8 (2001),21-28.[5] K. Yosida, Functional Analysis. Springer Verlag, Berlin-Heidelberg-New York (1974).[6] Nguyen Xuan Thao, On the generalized convolution for the Stieltjes,Hilbert, Fourier cosine and sine transforms (in Russian). Ukr. Mat. J.53, No 4 (2001), 560-567.[7] Nguyen Xuan Thao and Nguyen Thanh Hai, Convolution for inte-gral transforms and their applications (in Russian). Russian Academy,Moscow (1997).A GENERALIZED CONVOLUTION WITH . . . 337[8] Nguyen Xuan Thao, V.A. Kakichev and Vu Kim Tuan, On the gen-eralized convolution for Fourier cosine and sine transforms. East-WestJ. Math. 1, No 1 (1998), 85-90.[9] Nguyen Xuan Thao and Trinh Tuan, On the generalized convolutionfor I-transform. Acta Mathematica Vietnamica 28, No 2 (2003), 159-174.[10] I. N. Sneddon, Fourier Transform. McGray Hill, New York (1951).[11] M. Saigo and S. B. Yakubovich, On the theory of convolution integralsfor G-transforms. Fukuoka Univ. Sci. Report 21, No 2 (1991), 181-193.[12] S. B. Yakubovich, On a constructive method for construction of inte-gral convolutions. DAN BSSR 34, No 7 (1990), 588-591.[13] S. B. Yakubovich and A. I. Mosinski, An integral equation and a con-volution for a transform of Kontorovich-Lebedev type. Diff. Uravnenia29, No 7 (1993), 1272-1284.[14] H. J. Glaeske and Vu Kim Tuan, Some applications of the convolu-tion theorem of the Hilbert transform. Integral Transforms and SpecialFunctions 3, No 4 (1995), 263-268.[15] H. M. Srivastava and Vu Kim Tuan, A new convolution theorem forthe Stieltjes transform and its application to a class of singular integralequations. Arch. Math. 64, No 2 (1995), 144-149.[16] Vu Kim Tuan and M. Saigo, Convolution of Hankel transform andits application to an integral involving Bessel functions of first kind.Internat. J. Math. Math. Sci. 18, No 3 (1995), 545-550.1 Hanoi Water Resources University175 Tay Son, Dong Da, Hanoi, VIETNAMe-mail: thaoxbmai@yahoo.com2 Department of Mathematics Received: September 9, 2004University of West GeorgiaCarrollton, GA30118, USAe-mail: vu@westga.edu3 Hanoi University of Transport and CommunicationsLang Thuong, Dong Da, Hanoi, VIETNAM

A Generalized Convolution with a Weight Function for the Fourier Cosine and Sine Transforms

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A Generalized Convolution with a Weight Function for the Fourier Cosine and Sine Transforms

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