The Fourier Laplace Generalized Convolutions and Applications to Integral Equations

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1 Vietnam J Math 3 4: DOI.7/ The Fourier Laplae Generalized Convolution and Appliation to Integral Equation Nguyen Xuan Thao Trinh Tuan Le Xuan Huy Reeived: 4 Otober / Aepted: Augut 3 / Publihed online: 5 November 3 The Author 3. Thi artile i publihed with open ae at Springerlink.om Abtrat In thi paper we introdue two generalized onvolution for the Fourier oine, Fourier ine and Laplae integral tranform. Convolution propertie and their appliation to olving integral equation and ytem of integral equation are onidered. Keyword Fourier ine tranform Fourier oine tranform Laplae tranform Mathemati Subjet Claifiation 33C 44A35 45E 45J5 47A3 47B5 Introdution Convolution for integral tranform are tudied in the early year of the th entury, uh a onvolution for the Fourier tranform ee, 9, 3, the Laplae tranform ee,, 8, 3, 6 9, the Mellin tranform ee 8, 3, the Hilbert tranform ee, 3, the Fourier oine and ine tranform ee 5, 7, 3, 4, and o on. Thee onvolution have many important appliation in image proeing, partial differential equation, integral equation, invere heat problem ee 4, 8, 3, 5 8. N.X. Thao Shool of Applied Mathemati and Informati, Hanoi Univerity of Siene and Tehnology, Dai Co Viet, Hanoi, Vietnam thaonxbmai@yahoo.om T. Tuan Department of Mathemati, Eletri Power Univerity, 35 Hoang Quo Viet, Hanoi, Vietnam tuantrinhpa@yahoo.om L.X. Huy B Faulty of Bai Siene, Univerity of Eonomi and Tehnial Indutrie, 456 Minh Khai, Hanoi, Vietnam lxhuy@uneti.edu.vn

2 45 N.X. Thao et al. In 998, in 6 the author introdued the general method for defining a generalized onvolution with a weight funtion γ for three arbitrary integral tranform K,K and K 3, uh that the following fatorization identity hold: K f γ g y = γ yk f yk 3 gy. Thi idea ha opened up many new reearhe and new onvolution with intereting propertie appearing in 7, but o far there i only one onvolution for Laplae tranform defined a follow ee, 9: whih atifie the fatorization identity Here L denote the Laplae tranform f L g x x = fx tgtdt, x >, L f L g y = Lf ylgy. Lf y = fxe yx dx, y >. In thi paper, we introdue and tudy two new generalized onvolution with a weight funtion for the Fourier oine-laplae and Fourier ine-laplae tranform. We alo obtain ome norm inequalitie of thee onvolution and algebrai propertie of onvolution operator on L R + and L α,β p R +. In the lat etion, we apply thee onvolution to olve everal lae of integral equation a well a ytem of two integral equation. Well-known Convolution The onvolution of two funtion f and g for the Fourier oine tranform i of the following form ee 3: f F g x = fy gx + y + g x y dy, x >, π whih atifie the following fatorization identity: F f F g y = F f yf gy y >. Here F i the Fourier oine tranform F f y = fxo xy dx, y >. π The generalized onvolution for the Fourier ine and Fourier oine tranform of f and g i defined a follow ee 3: f g x = fu g x u gx + u du, x >, 3 π

3 The Fourier Laplae Generalized Convolution 453 whih atifie the following fatorization identity: F f g y = F f yf gy y >. 4 Here F i the Fourier ine tranform F f y = fxin xy dx, y >. π The onvolution of two funtion f and g with a weight funtion for the Fourier ine tranform i of the following form ee 5: f γ g x = F fy ignx + y g x + y π gx + y + + ignx y + g x y + ignx y g x y dy, x >, 5 whih atifie the fatorization equality F f γ g y = in yf f yf gy F y >. 6 The onvolution of two funtion f and g for the Fourier oine and Fourier ine tranform i of the following form ee 7: f g x = fy gx + y + igny xg y x dy, x >, 7 π whih atifie the following fatorization identity: F f g y = F f yf gy y >. In thi paper we are intereted in the weighted pae L α,β p R + L p R +,x α e βx dx with the norm defined a follow: fx α,β = L p R + fx /p p x α e dx βx, p<. 3 The Fourier Laplae Generalized Convolution Definition The generalized onvolution with a weight funtion γy= e μy,μ>of two funtion f and g for the Fourier oine-laplae and Fourier ine-laplae tranform are defined by γ f g { }x = π + x u where x>. ± + x + u fugvdudv, 8

4 454 N.X. Thao et al. Theorem For two arbitrary funtion fxand gx in L R +, the generalized onvolution f g { } belong to L R +. Moreover, the following norm etimate and fatorization identitie hold: f γ g L { f } R + L R + g L R +, γ F { } f g { }y = e μy F { }f ylgy y >. 9 Furthermore, the generalized onvolution f γ g { } belong to C R +. Proof We have = + x u ± + x + u dx u + t dt + u + t dt dt = π. + t From 8 and, we have f γ g { }x dx fu du gv dv = f L R + g L R +. Therefore Thu f γ g { } L R + f L R + g L R + <. f γ g { } L R +. From 8 and by applying formula e αx o xy dx = α α +y α > ee, we obtain f γ g { }x = π = π R 3 + R 3 + = π = π fugve v+μy ox uy ± ox + uy dudvdy { } o yx o yu fugve v+μy dudvdy in yx in yu { } o yu { } o xy fu du gve vy dv e μy dy in yu in xy { } o xy F { }f ylgye μy dy. in xy From and, we get the fatorization identitie 9. From and Riemann Lebegue lemma, we obtain f γ g { } C R +. Theorem i proved.

5 The Fourier Laplae Generalized Convolution 455 Theorem Suppoe that p>,r, <β,fx L p R +, gx L R +. Then the generalized onvolution f γ g { } are well-defined, ontinuou and belong to L α,β r R +. Moreover, we get the following etimate: f γ g L { } α,β C f r R + L pr + g L R +, 3 where C = r Γ /r α + and Γxi Gamma Euler funtion. Furthermore, if fx L R + L p R + then the generalized onvolution f γ g { } belong to C R +, and atify the fatorization identity 9. πμ /p β α+ Proof By applying Hölder inequality for q>, p + = and, we have q f γ g { } { π { π R + fu p + x u ± gv /p dudv} + x + u R + R + gv + x u ± + x + u fu p gv /p μ dudv /q gv πdv /p = f LpR+ g LR+. πμ } /q dudv Thu, onvolution 8 exit and are ontinuou. Combining with formula in, p. 5, we get x α e βx f γ g { }x r dx C r f r L pr + g r L R +. Hene onvolution 8 areinl α,β r R + and identitie 3 hold. From the hypothei of Theorem, and by imilar argument a in Theorem, we get the fatorization identitie 9. Combining with Riemann Lebegue lemma, we obtain f γ g { } x C R +. Theorem i proved. Theorem 3 Let α>, <β,p >,q >,r be uh that p + =. Then q for fx L p R + and gx L q R +, + x q, the onvolution f γ g { } are welldefined, ontinuou, bounded in L α,β r R + and f γ g L { } α,β C f r R + L pr + g Lq R +,+x q, 4 where C = μ p π q β α+ r Γ /r α +. Moreover, if fx L R + L p R + and gx L R + L q R +, + x q then onvolution f γ g { } belong to C R + and atify fatorization identitie 9.

6 456 N.X. Thao et al. Proof Applying Hölder inequality for p,q > and ombining with, we have f γ g { } { π { fu p + x u ± + x + u + v dudv gv q R + R + + x u ± + x + u q /q dudv} + v fu p /p du π μ + v dv gv q /q + v q πdv = μ p π q f LpR + g Lq R +,+x q. Therefore, the onvolution 8 are well-defined and ontinuou. From that and by applying formula in, p. 5, we obtain x α e βx f γ g { }x r dx C r f r L pr + g r L q R +,+x q. It how that the onvolution 8areinL α,β r R +, and etimate 4 hold. From hypothei of Theorem 3, by imilar argument a in Theorem, we get the fatorization identitie 9. Combining with the Riemann Lebegue lemma, we obtain f γ g { } x C R +. Theorem 3 i proved. Corollary Under the ame hypothei a in Theorem 3, the generalized onvolution 8 are well-defined, ontinuou, belong to L p R +, and the following inequalitie hold: } /p f γ g { } LpR + π /p f LpR + g Lq R +,+x q. 5 Furthermore, in the ae p =, we get the following Pareval identity: f γ g { }x dx = e μy F { }f ylgy dy. 6 Proof By applying Hölder inequality and, we have f γ g { }x p dx { fu p π p R + v + x u + p /p ± q + x + u dudv gv /q } p q dudv dx R + v +

7 The Fourier Laplae Generalized Convolution 457 fu p π p dudv + v q gv p/q dv q π p R + v + = + v dv fu p du + v q gv p/q dv q = π f p L pr + g p L q R +,+x q. Therefore, the onvolution f γ g { } x are ontinuou in L pr + and 5 hold. On the other hand, we get the following Pareval equalitie in L R + : F { }f L R + = f L R +. Combining with fatorization identitie 9, we get the Fourier-type Pareval identity 6. Corollary a Let fx L R +, gx L R +. Then the generalized onvolution 8 are welldefined in L α,β r R + r,β,α>, and the following etimate hold: f γ g L { } α,β r R + α+ πμ β r Γ /r α + f L R + g L R +. 7 b If fx,gx L R + then onvolution 8 are well-defined in L α,β r R + r, β,α> and the following etimate hold: Proof f γ g L { } α,β α+ r R + πμ β r Γ /r α + f L R + g L R +. 8 a By applying Shwarz inequality and, we have f γ g { }x π gv / dv fu gv / π R μ dudv + = πμ f L R + g L R +. Combining with formula in, p.5, we get f γ g L { } α,β r Thu, 7 i proved. R + α+ πμ β r Γ /r α + f L R + g L R +.

8 458 N.X. Thao et al. b By applying Shwarz inequality, we have f γ g { }x fu gv / π μ dudv R + = πμ f L R + g L R +. R + fu gv μ dudv / Combining with formula in, p.5, we get 8. Theorem 4 Tithmarh Type Theorem Given two ontinuou funtion g L R +, f L R +,e γx, γ >. If f γ g x = x > then either fx= x > or gx = x >. Proof We have d n o yxfx dy n = fxx n o yx + n π e γx x n e γx fx n! γ n e γx fx. 9 Here we ued the following etimate: e γx x n = e γxγ xn n! n! γ n e γx γx n! e γ = n! n γ, n and f L R +,e γx d. Combining with 9weget n dy n o yxfx L R +. Sine L R +,e γx L R +, F f y are analyti in R +. On the other hand, we find that Lgy i analyti in R +. By uing the fatorization propertie 9 forf γ g x = we have F f ylgy = y >. It implie that either fx= x >orgx = x >. Theorem 4 i proved. Corollary 3 Under the ame hypothei a in Theorem 4, if f γ g x = x > then either fx= x > or gx = x >. Propoition Let fxand gx be two funtion in L R +. Then f γ g { }x = π gv fu { F } + u Here, the onvolution F, are defined by, 3, repetively. x dv. Proof From 8, and3, we have γ f g { }x = π + x u ± + x + u fugvdudv

9 The Fourier Laplae Generalized Convolution 459 = π = π { gv fu gv + x u ± fu { F } + u x dv. + x + u } du dv Propoition Let fx,gxand hx be funtion in L R +. Then onvolution 8 are not ommutative and aoiative but atify the following equalitie: a f γ F g γ h = f γ F g γ h, b f F g γ h = f F g γ h, f g γ h = f g γ h, d f g γ h = f g γ h. Here the onvolution γ F, F, and are defined by 5,, 3 and 7, repetively. Proof From 6 and9, we have F f γ γ γ g h y = in yf F f yf g h y = e μy in yf f yf gylhy = e μy F f γ g ylhy = F f γ γ g h y. F F Hene f γ g γ h = f γ g γ h. F F The proof of b,, and d are imilar. 4 Integral Equation and Sytem of Integral Equation In thi etion we introdue everal lae of integral equation and ytem of two integral equation related to onvolution 8 whih an be olved in a loed form. a Conider integral equation of the firt kind where θ { } x, u = π ϕv θ { }x, uf u du = gx, x >, + x u ± + x + u dv, μ >. Put HR + ={h L R +, h = F { }f y}. We onider the retrition mapping F { } : HR + L R +.

10 46 N.X. Thao et al. Theorem 5 Let gx,ϕx L R + and uppoe that g x, g x be uh that gx = γ g g { }x. Then the neeary and uffiient ondition to enure that the equation have olution in L R + i that F { }g ylg y HR Lϕy +. Moreover, the olution are given in the following loed form: fx= F { }g ylg y Lϕy { } o xy dy. in xy Proof Neeity. By the hypothei, equation ha olution in L R + givenby. Sine gx L R + therefore f γ ϕ { } x L R +. From that, by applying the fatorization propertie 9 for, we have therefore e μy F { }f ylϕy = e μy F { }g ylg y, F { }f y = F { } g ylg y. Lϕy Sine F { }f y L R + hene F { }f y HR +. From that and we get F { } g ylg y HR Lϕy +. Suffiieny. By the hypothei F { }g ylg y HR Lϕy +, therefore there exit fx L R + atifying F { }f y = F { }g ylg y, hene Lϕy F { }f ylϕy = F { }g ylg y. Therefore γ f ϕ and we obtain. Theorem 5 i proved. { }x = gx, b Conider integral equation of the eond kind fx+ ftθ { }x, t dt = gx, x >, 3 where and θ { } x, u = R + H { } x,u,v= π π H { } x,u,v ψ u x ± ψu+ x ϕvdudv, + x u ± + x + u. 4 Theorem 6 Let ϕx,ψx L R +. Then the neeary and uffiient ondition to enure that the equation 3 have unique olution in L R + for all gx in L R + i that

11 The Fourier Laplae Generalized Convolution 46 + F ψ γ ϕ y y >. Moreover, the olution an be preented in loed form a follow: fx= gx g q x, 5 { F } where the onvolution, are defined by, 3, repetively, and q L R + i F defined by F qy = F ψ γ ϕ y + F ψ γ. 6 ϕ y Proof Neeity. We an rewrite equation 3 in the form fx+ f ψ γ ϕ { F } { }x = gx. 7 Aume that the integral equation 3 have unique olution in L R + for all g in L R +. Therefore, there exit g L R + uh that F { }gy y >. 8 By uing fatorization propertie 9,, and 4for7, we get F { }f y + e μy F { }f yf ψylϕy = F { }gy. Combining with 9, we obtain F { }f y + F ψ γ ϕ y = F { }gy. 9 Uing feedbak evidene, aume that there exit y > uh that + F ψ γ ϕ y =. Combining with 9, we get F { }gy = g L R +. It i a ontradition to 8. Hene + F ψ γ ϕ y y >. Suffiieny. From8 and the aumption of Theorem 6, wehave F { F { }f y = }gy + F ψ γ ϕ y = F { }gy F ψ γ ϕ y + F ψ γ ϕ y F ψ γ ϕ y = F { }gy F { }gy + F ψ γ. 3 ϕ y With the ondition + F ψ γ ϕ y y >, due to Wiener Levy theorem in 9, p. 63, there exit a funtion q L R + atifying 6. Combining with 3, we have F { }f y = F { }gy F { }gyf qy = F { }gy F { } g { F } q y.

12 46 N.X. Thao et al. Therefore we get 5. Theorem 6 i proved. We onider the ytem of two integral equation Here fx+ gx + gtmx,tdt = px, ftnx,tdt = qx, x >. Mx,t = H x,u,v k u x + ku + t ϕvdudv, R + Nx,t = H x,u,v l u x + lu + t ψvdudv R + 3 and H i defined by 4. Theorem 7 Suppoe that ϕx, ψx, px, qx L R + are uh that F k γ ϕ F l γ ψ y y >. Then ytem 3 ha a unique olution f, g in L R +, L R + given by formula x + p F ξ x γ q k ϕ F ξ F x, 3 fx= px γ q k ϕ F gx = qx γ p l ψ F x + q F ξ x γ p l ψ F Here ξ L R + i uh that F ξ x. 33 F k γ ϕ l γ ψ y F F ξy = F k γ ϕ l γ. 34 ψ y F Proof We an rewrite ytem of two equation 3 in the following form: fx+ g k γ ϕ x = px, F gx + f l γ ψ 35 x = qx. F By uing fatorization propertie 9, for35, we get Therefore F f y + e μy F gyf kylϕy = F py, F gy + e μy F f yf lylψy = F qy. F f y + F gyf k γ ϕ y = F py, F gy + F f yf l γ ψ y = F qy. 36

13 The Fourier Laplae Generalized Convolution 463 Solving the ytem of two linear equation 36, we get F py F q k γ ϕ y F F f y = F k γ ϕ l γ ψ y F = F py F q F k γ ϕ y γ F k ϕ l γ ψ y F + γ F k ϕ l γ. 37 ψ y F In virtue of Wiener Levy theorem, there exit a funtion ξ L R + atifying 34. Combining with 37, we have F f y = γ F py F q F k ϕ y + F ξy γ = F py F q F k ϕ y + F p ξy F F q F k γ ϕ F ξ y. Therefore we obtain 3. Similarly, we get 33. Theorem 7 i proved. We now onider the ytem 3 with Mx,t = H x,u,v k u x ku + t ϕvdudv, Nx,t = R + R + where H i defined by 4. H x,u,v l u x lu + t ψvdudv, Corollary 4 Under the ame hypothei a in Theorem 7, the ytem 3 ha unique olution f, g in L R +, L R + given by formula fx= px γ q k ϕ x + p ξ x γ q k ϕ ξ x, gx = qx γ p l ψ x + q ξ x γ p l ψ ξ x. Here ξ L R + i defined by 34. Aknowledgement Thi reearh i funded by Vietnam National Foundation for Siene and Tehnology Development NAFOSTED under grant number.-.5. The author would like to expre their deep gratitude to the referee for hi/her ontrutive omment. Open Ae Thi artile i ditributed under the term of the Creative Common Attribution Liene whih permit any ue, ditribution, and reprodution in any medium, provided the original author and the oure are redited.

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ON THE LAPLACE GENERALIZED CONVOLUTION TRANSFORM

Annale Univ. Si. Budapet., Set. Comp. 43 4) 33 36 ON THE LAPLACE GENERALIZED CONVOLUTION TRANSFORM Le Xuan Huy and Nguyen Xuan Thao Hanoi, Vietnam) Reeived February 5, 3; aepted Augut, 4) Abtrat. Several