CONSTRUCTIVE AND INVESTIGATE THE Title GENERALIZED CONVOLUTION WITH THE WE FUNCTION FOR THE FOURIER COSINE AND TRANSFORMS

Transcription

1 CONSTRUCTIVE AND INVESTIGATE THE Title GENERALIZED CONVOLUTION WITH THE WE FUNCTION FOR THE FOURIER COSINE AND TRANSFORMS Authr(s) Nguyen, Xuan Tha; Nguyen, Thanh H Annual Reprt f FY 2007, The Cre Citatin between Japan Sciety fr the Prm Vietnamese Academy f Science and T P.580-P.589 Issue 2008 Date Text Versin publisher URL DOI Rights Osaka University

2 Osaka University

3 CONSTRUCTIVE AND INVESTIGATE THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE FOURIER COSINE AND SINE TRANSFORMS NGUYEN XUAN THAO AND NGUYEN THANH HONG ABSTRACT. A generalized cnvlutin with the weight functin fr the Furier csine and sine transfrms is intrduced. It's prperties and applicatins t slve systems f integral equatins are cnsidered. Let Fs be the Furier sine transfrm [2] and Fe be the Furier csine transfrm [2]. INTRODUCTION (Fsf)(c) = [$ ] sinxyf(y)dy, (FcJ)(c) = [$ ] csxyf(y)dy. Cnvlutin thery has been studied in 20 th. Firstly, the cnvlutins fr the Furier; Laplace and Mellin transfrms have investigated. Later n, the cnvlutins fr the integral transfrms Hilbert, Hankel, Kntrvich - Lebedev and Stieltjes have already investigated. The cnvlutin f tw functins f and 9 fr the Furier csine transfrm is intrduced in [7] 00 (. ) U * g)(x) = ~ J Fe V 27f which satisfied the fllwing factrizatin equality (.2) FeU * g)(y) = (FcJ)(y)(Fcg)(y), Fe f(y)[g(lx - yl) + g(x + y)]dy, x> 0, Vy> O. In 958, Vilenkin LYa intrduced the first cnvlutin with the weight functin fr the transfrm Mehler - Fck. In 967, Kakichev V.A prpsed a cnstructive methd fr defining the cnvlutin with a weight functin fr an arbitrary integral transfrm (see [4]). He cnstructed the cnvlutin f tw functins f and 9 with the weight functin (y) = sin y fr the Furier Cnvlutin, Furier sine transfrm, Furier csine transfrm, integral equatin. 580-

4 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fc) Fs TRANSFORMS sine transfrm which is f the frm [4] and [0] +00 (.3) U : g)(x) = ~ J f(y) [sign(x + y - l)g(lx + y - ) + sign(x - y + l)g(lx - y + ) Fs 2y 27f - g(x + y + ) sign(x - y - l)g(lx - y - )]dy, x > 0, and prved the fllwing factrizatin identity [4], [0] (.4) FsU;' g)(y) = siny(fsf)(y)(fsg)(y), Vy> O. The cnvlutin with the weight functin ~f(y) functins f and g is intrduced in [] = cs y fr the Furier csine transfrm f tw +00 (.5) U~g)(x) = ~ J f(y)[g(ly+x-ll)+g(ly-x-ll)+g(y+x+l)+g(ly-x+ll)]dy,x > 0 Fc 2y 27f and satisfy the factrizatin equality [] (.6) FeU ~ g)(y) = csy(fcf)(y)(feg)(y), Vy> O. Fc In 94, Churchill R.V intrduced the first generalized cnvlutin f tw functins f and g fr the Furier sine and Furier csine transfrms [7] 00 (. 7) U * g)(x) = ~ J y27f and prved the fllwing factrizatin identity [7] f(u)[g(lx - ul) - g(x + u)]du, x> 0, (.8) FsU i g)(y) = (Fsf)(y) (Feg)(y), Vy > O. In the nineties f the last century, Yakubvich S. B has intrduced several generalized cnvlutins with index f the Mellin transfrm, Kntrvich-Lebedev transfrm, G-transfrm and H-transfrm. In 998, Kakichev and Nguyen Xuan Tha prpsed a cnstructive methd fr defining the generalized cnvlutin fr three arbitrary integral transfrms (see [5]). Up t nw, based n this methd, several new generalized cnvlutins fr the integral transfrms were established and investigated. The generalized cnvlutin f tw functins f and g fr the Furier csine and sine transfrms is defined by [6] 00 (.9) U * g)(x) = ~ J 2 y 27f f(u) [sign(u - x)g(lu xl) + g(u + x)]du, x> O. Fr this generalized cnvlutin the fllwing factrizatin equality hlds [6] (.0) FeU * g)(y) = (Fsf)(y)(Fsg)(y), Vy> O

5 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fc,Fs TRANSFORMS Anther generalized cnvlutin with the weight functin (y) = sin y fr the Furier csine and sine has been studied in [5] (. ) 00 ' U*g)(x)= ~ J f(u)[g(lx+u-ll)+g(lx-u+ll) g(x+u+l)-g(lx-u-ll)]du,x> O. 2 2v 2T It satisfies the factrizatin prperty [5] (.2) FeUi g)(y) siny(fsf)(y)(feg)(y), Vy> O. 2 The generalized cnvlutin f tw functins f and 9 with the weight functin (y) = sin y fr the Furier sine and csine transfrms has the frm (.3 ) 00 ~Il U * g)(x) J = ~ f(u)[g(lx+u -) +g(lx - U-ll) 2v 2T and satisfy the fllwing factrizatin identity [4] (.4) FsU i g)(y) = siny(fcj)(y)(fcg)(y), Vy> O. g(x +u+ ) g(lx u+ II)]du, x> 0, In this paper we cnstruct a new generalized cnvlutin with a weight functin fr the Furier csine and sine transfrms. Is prperties and the relatin with several well-knwn cnvlutins and generalized cnvlutins are cnsidered. We als apply this ntin t slve a system f integral equatins. 2. THB GBNBRALIZBD CONVOLUTION Definitin. A generalized cnvlutin with the weight functin I(Y) = cs Y fr the Furier csine and sine transfrms f functins f and 9 is defined by +00 (2.) U i g)(x) j' ~ f(y) [g(y + x + ) + sign(y x l)g(ly x ) 2v 2T + sign(y + x - l)g(ly + x ) + sign(y - x + l)g(ly - x + )]dy, x> O. We dente by L(lR.+) the space f all functins f defined n such that J + If(x)ldx < 00. Therem. Let f and 9 be functins in L(lR.+) then the generalized cnvlzdin U i g) (x) defined by (2.) als be a L(lR.+) functin. Mrever, the fllwing factrizatin equality hlds (2.2) FeU i g)(y) csy(fsj)(y)(fsg)(y), Vy > O

6 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fc,Fs TRANSFORMS Prf. Frm the defining frmula f the generalized cnvlutin and the fact that f, 9 E L(lR+) we have (2.3) +00 IU g)l(x)dx = ~ J J J If(y)I I[g(y + x + ) + sign(y - x - l)g(ly x-ii) 2y 2f sign(y + x - l)g(ly + x-ii) + sign(y - x + l)g(ly - x + )]ldydx +00 J If(y)l [J Ig(y + x + )ldx + J Ig(ly - x - )ldx On the ther hand, J Ig(ly+x-ll)ldx+ J Ig(ly x+ll)ldx]dy. 0 (2.4) J Ig(y + x + l)ldx + J Ig(ly x )ldx = J Ig(t)ldt + J Ig(ltl)ldt Similary, 0 y+l -y-l +00 = 2 J Ig(t)ldt. +00 (2.5) J Ig(ly + x - )Idx + J Ig(ly - x + )Idx = 2 J Ig(t)ldt. 0 0 Frm (2.3), (2.4) and (2.5) ne hlds S U g)l(x) belng t L(lR+). +00 J IU g)l(x)dx ~./r J Ig(t)ldt J Ig(t)ldt ~ Nw we prve the factrizatin equality (2.2). We have cs y(fsj) (y)(fsg) (y) = ~ J J csy sinuy sinvyf(u)g(v)dudv 0 =2~ J J f(u)g(v)[csy(u v + ) + csy(u - v-i) - csy(u + v + ) - csy(u + v - l)]dudv

7 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fc,Fs TRANSFORMS Changing the variables gives +00 csy(fsf)(y)(fsg)(y) = 2~ j f(u) [ j g(t + u + ) csytdt + j g(t + u - ) csytdt -u-l -u+l - j g(t-u-l)csytdt- j g(t-u+l)cosytdt]du u+l +00 =2~ j f(u)[j g(u+t+l)csytdt+ j sign(u-t+l)g(lu t+ll)csytdt j sign(u + t - l)g(lu + t - ) csytdt + j sign(u - t -)g(lu - t - ) cs ytdt] du 0 =~ j j f(u) [g(u + t + ) + sign(u - t + l)g(lu - t + ) + sign(u + t - l)g(lu + t - ) 27f 0 + sign(u - t - l)g(lu - t - ll)]ducsytdt =/! JOO(J i g)(t) csytdt =Fe(J i g)(y). 'u-l It shws that Fe(J i g)(y) = csy(fsf)(y)(fsg)(y). The prf f Therem is cmpletes. Therem 2. In the space L(lR+) the generalized cnvlutin (2.) is nt assciative and the relatin with knwns cnvlutins and generalized cnvlutins as fllws a) f * (g ~ h) g * (J ~ h) = h * (g ~ ), where f * g is defined by (. 7). b) f ~ (g * h) = (J ~ g) * h = (J * h) i g, where f * g is the Furier csine cnvlutin (.). ~ ~ -ll'y 'Yl'Y 'Yl'Y 'Yl 'Yl ) () c) f * (g*h) = (J * h) *g = h * (J*g), where f * g and f * g are defined by (.3 and.. 2 Fs 2 Fs 2 d) f i (g i h) = (J i g) ~ h = (J i h) i g, where fig is the Furier csine cnvlutin with 2 Fe 2 Fe Fc a weight functin (.5)

8 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fc,Fs TRANSFORMS Prf. a) Frm the factrizatin equality, we have f f FsU * (g * h))(y) =(FsJ)(y)(Fc(g * h))(y) =(FsJ)(y) csy(fsg)(y)(fsh)(y) = (Fsg)(y) csy(fsj)(y)(fsh)(y) =(Fsg)(y)(FeU h))(y) =Fs(g * U h))(y). Which implies that f * (g h) = g * U h) On the ther hand, FsU * (g h))(y) =(FsJ)(y) csy(fsg)(y)(fsh)(y) =(Fsh)(y) csy(fsg)(y)(fsf)(y) f =Fs(g * U * h))(y). Then we btain the part a). The parts b), c), d) can be btain in similar way. The Therem is prved. [] Therem 3. In the space L(R+) the generalized cnvlutin (2.) des nt have a unit element. Prf. Suppse that there exists a unit element e f the generalized cnvlutin (2.) in L(lR+). It means that f e = e f = f fr any functin f E L(lR+). It fllws that FeU e) (y) = (Fcf) (y), \ly> O. Hence, csy(fse)(y)(fsj)(y) = (Fcf)(y), \ly > O. Chsing f(x) = e- x E L(lR+). Frm the fact that we btain (Fse)(y) = --. ycsy It is cntraditin frm the fact that -- :. L(lR+) while (Fse)(y) E L(lR+) since e E L(lR+). ycsy The Therem is prved. [] Let L(lR+, ex) = {h, fr all exh(x) E L(lR+)}. Therem 4 (Titchmarch-type Therem). Let f, g E L(lR+, ex). If U g)(x) == 0 then either f == 0 r g == O. Prf. Suppse that U * g)(x) = 0, \Ix> 0, in view f Therem c;s (2.6) FeU g)(y) = csy(fsj)(y)(f~g)(y) = 0, \ly > O

9 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fe, Fs TRANSFORMS We have Id~n[sin(XY)f(x)] = If(x)xnSin(xy+n~) where h (x) ex f(x) E L(lP?+). ~ If(x)Xnl le-x,xnl,lh(x) ~ n!lh(x)i, Due t \i\teierstrass criterin, the integral jdd n [sin(xy)f(x)]dx unifrmly cnverges n le.+. yn Therefre, based n the differentiability f integrals depending n parameter, we cnclude that (Fsf)(y) is analytic. Similarly, (Fsg)(y) is analytic. S frm (2.6) we have (Fsf)(y) == 0 r (Fsg) (y) == O. It cmpletes the prf APPLICATION TO SOLVE SYSTEMS OF INTEGRAL EQUATIONS Cnsider a system f integral equatins (3.) +00 f(x) + Al J g(t)cpi(x, t)dt + A2 J g(t)]i(x, t)dt + A3 J g(t)!li(x, t)dt = h(x) 0 0 A4 vh J ~(t)[f(x + t) - f(lx - tl)]dt + A5 J g(t)/jl (x, t)dt + g(.r) = k(x), 0 CPI(X, t) = ~[cp(t + x + ) + sign(t - x - l)cp(lt - x-ii) + sign(t + x - l)cp(lt + x-ii) 2v 27T + sign(t x + l)cp(lt - x + )], ](X, t) = ;;:;=:[](lx + t - ) + ](lx - t + ) -](x + t + ) -](lx - t - )], 2v 27T.!LI(X, t) = ;;:;=:[slgn(t - x)!i(lt - xl) +!i(t + x)], V 27T /Jl(X, t) = 2V'ST[/J(lx + t - ) + /J(lx - t - ) -/J(x + t + ) /J(lx - t + )], and /J ( x) = (VI * V2) ( x). 2 It shws that CPI, ]:!LI, /J and /JI are als L(lE.+) - functins Therem 5. With the cnditin -586

10 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fc,Fs TRANSFORMS there exists a unique slutin in L(lR+) f system (3.) which is f the frm f =h }..lcp ~ k - }..2k ~ ] }..3/* k + h * l - }..l(cp ~ k) * l- }..2(k -~ ]) * l - }..3(--* k) * l 2 2 Fe Fe 2 Fe 2 Fe n ~ 9 = k - }..4 ~ * h - }..5 W * h + k * l - }..4 (~ * h) * l - }..5 (W * h) * l Here l E L(lR+) and is defined by Fel Prf. System (3.) can be rewriten in the frm (3.2) f(x) + }..l(g ~ cp)(x) + }..2(g ~ ]) (x) + }..3(g * {L)(x) = h(x) 2 2 }..4(~ i f)(x) + }..5(f t j))(x) + g(x) = k(x). Using the respectively factrizatin equalities f abve generalized cnvlutins we have (3.3) (Fcf)(y) +}.. csy(fsg)(y)(fscp)(y) +}..2 sin y(fsg)(y) (Fc])(y) + }..3(Fsg) (y)(fsl-l) (y) = (Feh)(y) 'vve have 6.= }..4(FsE,) (y) (Fcf) (y) +}..5 siny(fcf)(y)(few)(y) + (Fsg)(y) = (Fsk)(y). }.. cs y(fscp) (y) +}..2 sin y(fe]) (y) + }..3(Fs{L)(y) }..4(Fs~)(Y) +}..5 siny(few)(y) = }..}..4 cs y(fscp) (y) (Fs~) (y) - }..2}..4 sin y(fc]) (y) (Fst;) (y) - }..3}..4 (Fsl-l) (y )}..4 (Fst;) (y)- }..}..5 sin y cs y(fscp) (y )(FeW) (y) }..2}..5 sin 2 y( Fc]) (y) (FeW) (y) - }..3}..5 sin y( Fsl-l) (y) (Fc/J) (y) I ')' I - - = - Fe ( }..}..4CP * ~ + }..2}..4t; * ] + }..3}..Ml * ~ + }..}..5CP * (VI * V2) + }..')}..5 Vl * (V2 * ]) }..3}..5Jl ~ W)(y) -=I- O. Then due t Wiener-Levi's Therem [0] there exists a functin l E L(lR+) such that 6. -= -587

11 furthermre.0. = THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fe,F. TRANSFORMS (Fch)(y) ) csy(fsip)(y) + A2 siny(fcrj)(y) + A3(FsfJ)(y) (Fsk)(y) =(Fch)(y) - Al csy(fsip)(y)(fsk)(y) A2 siny(fcrj)(y)(fsk)(y) - A3(FsfJ)(y)(Fsk)(y) Therefre (Fsf)(y) It shws that I ' I f = h - Alip * k - A2k * rj - A3fJ * k + h * l - Al(ip * k) * l - A2(k * rj) * l - A3(fJ * k) * l 2 2 Fe Fe 2 Fe 2 Fe Similarlly.0.= Then.0. 2 ( ) (Fsg)(y) =A = Fs k - A4~ * h - A5/J * h (y)[l + (Fel)(y)] w. A4(F.90(y) + A5 sin y(fc/j) (y) =(Fsk)(y) A4(Fs~)(y)(Fch)(y) =Fs(k - A4~ * h - A5/J i h)(y). (Fch)(y) (Fsk)(y) A5 sin y( Fc/J )(y) (Feh) (y) Hence g = k - A4~ * h The prf is cmplete. REFERENCES [] Bateman H. and Erdelyi A. (954),, Newyrk - Trnt - Lndn- McGraw-Hill Bk cmpany, INC. [2] Bchner S. and Chandrasekharan K. (949), Princetn Univ. Press, Princetn. [3] Churchill R. V. (94) New Yrk, 58p. [4] Kakichev V. A.(967), "On the Cnvlutin fr Integral Transfrms." N. 2, (In Russian). [5] Kakichev V. A and Nguyen Xuan Tha (998), "On the design methd fr the Generalized Integral Cnvlutins." N.,3-40 (In Russian). [6] Kakichev V.A, Nguyen Xuan Tha and Vu Kim Tuan (998), "On the Generalized Cnvlutins fr Furier Csine and Sine Transfrms." Vl. I, N. pp [7] LN. Sneddn (972), McGray-Hill, New Yrk

12 THE GENERALIZED CONVOLUTION WITH THE WEIGHT FUNCTION FOR THE Fc,Fs TRANSFORMS [8] Stein E.IvI. and Weiss G (97), Princetn Univ. Press, Princetn. [9] Nguyen Xuan Tha (200), "On the Generalized Cnvlutin fr the Stieltjes, Hilbert, Furier Csine and Sine Transfrms." V!' 53, p (in Russian). [0] Nguyen Xuan Tha, Nguyen Thanh Hai (997), " Cnvlutins fr Integral Transfrm and their Applicatin." Mscw, 44pages (Russian). [] Nguyen Xuan Tha and Nguyen Minh Kha (2004), "On the Cnvlutin with a weight-functin fr the Csine-Furier integral transfrm." 29, p [2] Nguyen Xuan Tha and Nguyen Minh Kha (2004), "Generalized Cnvlutin fr Integral Transfrms." Methds f cmplex and cliffrd Analysis. Prc. Inter. Cnf. App!. Math. SAS Inter. Public. p [3] Nguyen Xu an Tha and Nguyen Minh Kha (2005), "On the Generalized Cnvlutin with a Weight- Functin fr Furier, Furier Csine and Sine Transfrms." V!' 33, N. 4, [4] Nguyen Xu an Tha, Nguyen Minh Kha (2006), "On the Generalized Cnvlutin with a Weight-Functin fr the Furier Sine and Csine Transfrms", Vl 7, N 9, [5] Nguyen Xuan Tha, Vu Kim Tuan, Nguyen Minh Kha (2004), "On the Generalized Cnvlutin with a 'Weight-Functin fr the Furier Csine and Sine Transfrms." Vl 7, n. 3, [6] Titchmarsh H. M.(967), 2nd Ed. Clarendn Press, Oxfrd. address:thanxbmai<llyah.cm.hngdhsp@yah.cm Hani Water Resurces University, 75 Tay Sn, Dng Da, Hani, Vietnam, -589-