Tính chập suy rộng của các phép biến đổi tích phân Fourier consine và kontorovich-lebedev với hàm trọng - ThS. Trịnh Tuân

TÝnh chËp suy réng cña c¸c phÐp biÕn ®æi tÝch ph©n Fourier<br /> consine vµ kontorovich – lebedev víi hµm träng<br /> Ths. TrÞnh Tu©n<br /> Bé m«n gi¶i tÝch- Tr­êng §¹i häc Thuû lîi<br /> Tãm t¾t: X©y dùng tÝnh chËp suy réng cña c¸c phÐp biÕn ®æi tÝch ph©n Fourier consine vµ<br /> kontorovich – lebedev tõ ®ã t×m hiÓu c¸c tÝnh chÊt cña nã. øng dông tÝch chËp nµy vµo gi¶i hÖ<br /> ph­¬ng tr×nh tÝch ph©n.<br /> <br /> 1. §Æt vÊn ®Ò<br /> Vµo ®Çu thÕ kû 19 ng­êi ta ®· x©y dùng ®­îc tÝch chËp cña c¸c phÐp biÕn ®æi tÝch ph©n<br /> Fourier vµ råi tiÕp ®Õn tÝch chËp cña c¸c phÐp biÕn ®æi tÝch ph©n lÇn l­ît ®­îc x©y dùng ®ã lµ<br /> phÐp biÕn ®æi tÝch ph©n Lapace, Mellin Hilber [12], Hankel [4, 13] vµ Stieltfes [11].<br /> Ch¼ng h¹n tÝch chËp cña phÐp biÕn ®æi tÝch ph©n Fourier [12].<br /> +1<br /> Z<br /> 1<br /> (f ¤ g)(x) = p f (x ¡ y)g(y)dy; x > 0; (1)<br /> F 2¼<br /> ¡ 1<br /> <br /> Tho¶ m·n ®¼ng thøc nh©n tö ho¸:<br /> <br /> F (f ¤ g)(y) = (F f )(y)(F g)(y); 8y 2 R:<br /> F<br /> <br /> Víi F lµ phÐp biÕn ®æi tÝch ph©n Fourier [1].<br /> N¨m 1941 Churchill R.V x©y dùng ®­îc tÝch chËp ®èi víi phÐp biÕn ®æi tÝch ph©n<br /> Fourier cosine [3].<br /> +1<br /> Z<br /> 1 £ ¤<br /> (f ¤ g)(x) = p f (u) g(x + u) + g(jx ¡ uj) du; x > 0; (2)<br /> Fc 2¼<br /> 0<br /> <br /> Tho¶ m·n ®¼ng thøc nh©n tù ho¸<br /> F c (f ¤ g)(y) = (F c f )(y)(F c g)(y); 8y > 0:<br /> Fc<br /> <br /> Víi Fc lµ phÐp biÕn ®æi tÝch ph©n Fourier cosine [1].<br /> Vµo nh÷ng n¨m 1967, 1987 tÝch chËp cña phÐp biÕn ®æi tÝch ph©n Kontorovich – lebedev<br /> ®èi víi hai hµm f, g ®· ®­îc Karichev V.A vµ Yakubovich S.B x©y dùng [4, 14].<br /> +1 Z<br /> Z +1<br /> 1 h 1 ³ xu xv uv ´ i<br /> (f ¤ g)(x) := exp ¡ + + f (u)g(v)dudv; x > 0 (3)<br /> 2¼ 2 v u x<br /> 0 0<br /> tho¶ m·n ®¼ng thøc nh©n tù ho¸<br /> K (f ¤ g)(y) = (K f )(y)(K g)(y); 8y > 0:<br /> Víi K lµ phÐp biÕn ®æi tÝch ph©n Kontorovich – lebedev [1].<br /> Nh÷ng n¨m 90 Yakubovich S.B ®· x©y dùng ®­îc mét sè tÝch chËp suy réng theo chØ sè<br /> cña c¸c phÐp biÕn ®æi tÝch ph©n Mellin [15], Kontorovich – Lebedev [16], G phÐp biÕn ®æi [9]<br /> vµ H phÐp biÕn ®æi [17].<br /> N¨m 1998 Kakichev V.A vµ NguyÔn Xu©n Th¶o ®· c«ng bè ®­îc ph­¬ng ph¸p kiÕn<br /> thiÕt tÝch chËp suy réng cña ba phÐp biÕn ®æi tÝch ph©n bÊt kú víi hµm träng [5] vµ ®· cã<br /> nh÷ng vÝ dô cô thÓ nhê ph­¬ng ph¸p nµy [6, 7, 8].<br /> 1<br />  Trong bµi b¸o nµy nhê [5] t¸c gi¶ x©y dùng tÝch chËp suy réng cña hai phÐp biÕn ®æi tÝch<br /> ph©n Fourier consine vµ Kontorovich – lebedev víi hµm träng tõ ®ã t×m hiÓu c¸c tÝch chËp cña<br /> nã. §Æc biÖt øng dông tÝch chËp nµy vµo gi¶i hÖ ph­¬ng tr×nh tÝch ph©n.<br /> <br /> 2. Néi dung<br /> §Þnh nghÜa: TÝch chËp suy réng cña c¸c hµm f, g víi hµm träng ° (y) = 1 ®èi víi c¸c<br /> ysh(¼y)<br /> phÐp biÕn ®æi tÝch ph©n Fourier cosine vµ Kontorovich – Lebedev:<br /> ebedev t ransforms<br /> +1 Z<br /> Z +1<br /> ° 1 1£ ¡ ¤<br /> (f ¤ g)(x) = 2 e u ch ( x + v)<br /> + e¡ u ch ( x ¡ v )<br /> f (u)g(v)dudv; x > 0 (4)<br /> ¼ u<br /> 0 0<br /> à ! à !<br /> 1 1 °<br /> §Þnh lý 1. Gi¶ sö<br /> f 2L ; R+ ; g2 L ; R+ , th× tÝch chËp (f ¤g)(x) 2 L ( R+ ) vµ tho¶ m·n<br /> x shx<br /> ®¼ng thøc nh©n tù ho¸:<br /> <br /> °<br /> F c (f ¤g)(y) = ° (y)(K ¡ 1 f )(y)(Fc g)(y); 8y > 0; (5)<br /> -1<br /> víi K lµ phÐp biÕn ®æi ng­îc cña phÐp biÕn ®æi Kontorovich – Lebedev.<br /> <br /> Chøng minh ®Þnh lý: Do u:sh(x + v)e¡ u:ch( x + v)<br /> ! 0 khi u; v ! + 1 ,do ®ã chóng ta cã:<br /> +1 Z<br /> +1<br /> ¯Z 1£ ¡ ¤ ¯<br /> ¯ ¯<br /> ¯ e u ch( x + v )<br /> + e¡ u ch ( x ¡ v )<br /> f (u)g(v)dudv¯<br /> u<br /> 0 0<br /> +1 Z<br /> Z +1<br /> 1¯¯e¡ u ch ( x ¡ v) ¯<br /> ¯<br /> 6 u ch ( x + v )<br /> + e¡ jf (u)j jg(v)jdudv<br /> u<br /> 0 0<br /> +1 Z<br /> Z +1<br /> 1 jg(v)j<br /> 6 jsh(x + v)j je¡ u ch ( x + v )<br /> j jf (u)j dudv<br /> u jsh(x + v)j<br /> 0 0<br /> +1 Z<br /> Z +1<br /> 1 jg(v)j<br /> + jsh(x ¡ v)j je¡ u ch ( x ¡ v )<br /> j jf (u)j dudv<br /> u jsh(x ¡ v)j<br /> 0 0<br /> <br /> +1 Z<br /> Z +1<br /> 6 C1 1 1<br /> jf (u)j jg(v)jdudv<br /> u jsh(x + v)j<br /> 0 0<br /> +1 Z<br /> Z +1<br /> 1 1<br /> + C2 jf (u)j jg(v)jdudv < + 1<br /> u jsh(x ¡ v)j<br /> 0 0<br /> <br /> víi C1, C2 lµ c¸c h»ng sè.<br /> MÆt kh¸c:<br /> +1<br /> Z<br /> 1 ¡<br /> sh(x + ®) e¡ u :ch( x + ®)<br /> dx = e u :ch ®<br /> (6)<br /> u<br /> 0<br /> <br /> vµ<br /> +1<br /> Z<br /> e¡ u e¡ u :ch v<br /> jsh(x ¡ v)j e¡ u ch ( x ¡ v)<br /> dx = 2 ¡ : (7)<br /> u u<br /> 0<br /> <br /> <br /> 2<br />  Tõ (6) nhËn ®­îc:<br /> +1 Z<br /> Z +1 Z<br /> +1<br /> <br /> e¡ u ch ( x + v )<br /> jf (u)j jg(v)jdudvdx =<br /> 0 0 0<br /> +1 Z<br /> Z +1 Z+1<br /> jf (u)j jg(v)j<br /> = jsh(x + v)je¡ u ch ( x + v)<br /> dudvdx<br /> sh(x + v)<br /> 0 0 0<br /> Z Z Z1<br /> + 1 + 1 +<br /> jf (u)j jg(v)j<br /> 6 jsh(x + v)je¡ u ch ( x + v)<br /> dudvdx<br /> shv<br /> 0 0 0<br /> 1 1<br /> Z Z ¡ u ch v<br /> + +<br /> e jg(v)j<br /> = jf (u)j dudv<br /> u jshvj<br /> 0 0<br /> +1 Z<br /> Z +1<br /> 6 jf (u)j jg(v)j<br /> dudv (8)<br /> u jshvj<br /> 0 0<br /> <br /> Tõ (7) chóng ta cã<br /> +1 Z<br /> Z +1 Z<br /> +1<br /> <br /> e¡ u ch ( x ¡ v)<br /> jf (u)j jg(v)jdudvdx =<br /> 0 0 0<br /> + 1<br /> Z Z Z + 1 +1<br /> jf (u)j jg(v)j<br /> = jsh(x ¡ v)je¡ u ch( x ¡ v )<br /> dudvdx<br /> jsh(x ¡ v)j<br /> 0 0 0<br /> +1 Z<br /> Z +1 Z<br /> +1<br /> <br /> = jsh(x ¡ v)je¡ u ch( x ¡ v )<br /> jf (u)j jg1 (v)jdudvdx<br /> 0 0 0<br /> +1 Z<br /> Z +1<br /> ³ 2 1 ¡ ´<br /> 6 e¡ u<br /> + e u ch v<br /> jf (u)jjg1 (v)jdudv<br /> u u<br /> 0 0<br /> +1 Z<br /> Z +1<br /> jf (u)j<br /> = 3 jg1 (v)jdudv < + 1 : (9)<br /> u<br /> 0 0<br /> <br /> Tõ (4), (8) vµ (9) ta cã:<br /> +1<br /> Z +1 Z<br /> Z +1 Z<br /> +1<br /> ¯ ° ¯ 1£ ¡ ¤<br /> ¯(f ¤g)(x) ¯dx 6 1 e u ch ( x + v )<br /> + e¡ uch( x ¡ v )<br /> jf (u)j jg(v)jdudvdx < + 1 :<br /> ¼2 u<br /> 0 0 0 0<br /> °<br /> Nh­ vËy chøng tá r»ng (f ¤g)(x) 2 L ( R+ ) .<br /> MÆt kh¸c<br /> q<br /> 2 +1 Z<br /> Z +1<br /> ¡ 1 ¼ 1³ 2 ´<br /> ° (y)(K f )(y)(Fc g)(y) = y cos(yv)sh(¼y)K i y (u)f (u)g(v)dudv<br /> ysh(¼y) u ¼2<br /> 0 0<br /> r 1<br /> Z Z<br /> + + 1<br /> 2 2 1<br /> = cos(yv)K i y (u)f (u)g(v)dudv:<br /> ¼2 ¼ u<br /> 0 0<br /> <br /> Tõ c«ng thøc 1 ([1] trang 130) th×<br /> <br /> 3<br />  ° (y)(K ¡ 1 f )(y)(F c g)(y) =<br /> r +1 Z<br /> Z +1 +1<br /> hZ i<br /> 2 2 1<br /> = 2 cos(yv)f (u)g(v) cos(y®) e¡ u ch ®<br /> d® dudv =<br /> ¼ ¼ u<br /> 0 0 0<br /> r Z Z1<br /> + 1 +<br /> n +1<br /> Z o<br /> 1 2 1 £ ¤<br /> = f (u)g(v) cosy(® + v) + cosy(® ¡ v) e¡ u ch®<br /> d® dudv<br /> ¼2 ¼ u<br /> 0 0 0 (10)<br /> MÆt kh¸c:<br /> +1<br /> Z +1<br /> Z<br /> cosy(® + v) e¡ u ch ®<br /> d® = cos(yt) e¡ u ch ( t ¡ v)<br /> dt (11)<br /> 0 v<br /> Z1<br /> + +1<br /> Z<br /> cosy(® ¡ v) e¡ u ch ®<br /> d® = cos(yt) e¡ u ch ( t + v)<br /> dt (12)<br /> 0 ¡ v<br /> <br /> Tõ (11) vµ (12) chóng ta cã:<br /> +1<br /> Z<br /> £ ¤<br /> cosy(® + v) + cosy(® ¡ v) e¡ u ch ®<br /> d® =<br /> 0<br /> Z1<br /> + Z0 +1<br /> Z<br /> ¡ u ch ( t ¡ v) ¡ u ch( t + v)<br /> = cos(yt) e dt + cos(yt) e dt + cos(yt) e¡ u ch ( t + v )<br /> dt<br /> v ¡ v 0<br /> <br /> +1<br /> Z +1<br /> Z Zv<br /> ¡ u ch ( t + v ) ¡ u ch( t v )<br /> = cos(yt) e dt + cos(yt) e dt + cos(yt) e¡ u ch ( t ¡ v )<br /> dt<br /> 0 v 0<br /> +1<br /> Z<br /> £ ¤<br /> = cos(yt) e¡ u ch( t + v)<br /> + e¡ u ch ( t ¡ v )<br /> dt (13)<br /> 0<br /> <br /> Tõ (10), (11), (12) vµ (13) th×:<br /> ° (y)(K ¡ 1 f )(y)(Fc g)(y) =<br /> r +1 Z<br /> Z +1 +1<br /> n Z<br /> 1 2 1 £ ¤ o<br /> = 2 f (u)g(v) cos(yt) e¡ u ch ( t + v )<br /> + e¡ u ch ( t ¡ v )<br /> dt dudv<br /> ¼ ¼ u<br /> 0 0 0<br /> r +1<br /> Z n 1 +1 Z<br /> Z +1<br /> o<br /> 2 1 £ ¡ u ch ( t ¡ v ) ¤<br /> = cos(yt) 2 f (u)g(v) e¡ u ch ( t + v )<br /> + e dudv dt<br /> ¼ ¼ u<br /> 0 0 0<br /> °<br /> = Fc (f ¤ g)(y):<br /> Nh­ vËy ®¼ng thøc nh©n tù ho¸ (5) ®­îc chøng minh.<br /> °<br /> §Þnh lý 2. TÝch chËp (f ¤g)(<br /> kh«ng cã tÝnh chÊt giao ho¸n vµ kÕt hîp nh­ng cã c¸c ®¼ng thøc<br /> sau ®©y:<br /> <br /> <br /> <br /> <br /> 4<br /> ng equalities<br /> ° °<br /> a) f ¤(g ¤ h) = (f ¤ g) ¤ h<br /> Fc Fc<br /> ° °<br /> b) f ¤ (g ¤ h) = (g¤f ) ¤ h<br /> Fc Fc<br /> ° ° ° °<br /> c) f ¤(g ¤ h) = g ¤ (f ¤ h)<br /> §Ó chøng minh ®Þnh lý nµy chóng ta sö dông §Þnh lý 1 vµ ®¼ng thøc nh©n tù ho¸ (2)<br /> °<br /> §Þnh lý 3. TÝch chËp (f ¤g)( kh«ng tån t¹i phÇn tö ®¬n vÞ.<br /> Sö dông ®Þnh lý 1 vµ c«ng thøc 9.7.4 ([2] trang 199)<br /> °<br /> §Þnh lý 4. TÝch chËp (f ¤g)( ®­îc biÓu thÞ nh­ sau:<br /> p +1<br /> Z<br /> ° 2 1<br /> (f ¤ g)(x) = p f (u)(e¡ u ch v<br /> ¤ gjvj)(x)du<br /> ¼ ¼ u F<br /> 0<br /> Sö dông (4) vµ (1) ®Ó chøng minh ®Þnh lý nµy.<br /> <br /> HÖ qu¶. Víi g lµ hµm ch½n th×<br /> p +1<br /> Z<br /> ° 2 1 ¡ ¢<br /> (f ¤ g)(x) = p f (u) e¡ u ch v<br /> ¤ g(v) (x)du:<br /> ¼ ¼ u Fc<br /> 0<br /> <br /> 3. óng dông gi¶i hÖ ph­¬ng tr×nh tÝch ph©n<br /> XÐt hÖ:<br /> 8 +1<br /> Z<br /> ><br /> ><br /> ><br /> > f (x) + ¸ 1 µ(x; u)g(u)du = h(x); x> 0<br /> ><br /> ><br /> <<br /> 0<br /> +1<br /> (15)<br /> ><br /> > Z<br /> ><br /> > £ ¤<br /> ><br /> > ¸ 2 p 1<br /> f (u) Ã(x + u) + Ã(jx ¡ uj) du + g(x) = k(x)<br /> : 2¼<br /> 0<br /> <br /> trong ®ã c¸c hµm ®· biÕt Ã; h; k 2 L (R+ ), vµ ' 2 L ( x1 ; R + ).<br /> +1<br /> Z unknowm func<br /> 1 ¡ u ch (x + v )<br /> µ(x; u) = [e + e¡ u ch( x ¡ v)<br /> ]' (v)dv<br /> 4¼<br /> 0<br /> <br /> 1 , 2 lµ c¸c h»ng sè phøc vµ f, g lµ c¸c Èn hµm<br /> <br /> §Þnh lý 5. NÕu cã ®iÒu kiÖn<br /> °<br /> 1 ¡ ¸ 1 ¸ 2 F c (' ¤ Ã)(y) 6<br /> = 0; 8y > 0<br /> th× hÖ (15) cã nghiÖm:<br /> ° °<br /> f (x) = h(x) + (l ¤ h)(x) ¡ ¸ 1 (' ¤ k)(x) ¡ ¸ 1 (l ¤ (' ¤ k))(x) 2 L (R + )<br /> Fc Fc<br /> <br /> g(x) = k(x) + (l ¤ k)(x) ¡ ¸ 2 (h ¤ Ã)(x) ¡ ¸ 2 (l ¤ (h ¤ Ã))(x) 2 L (R + )<br /> Fc Fc Fc Fc<br /> <br /> trong ®ã l(x) 2 L (R+ ) vµ tho¶ m·n<br /> °<br /> ¸ 1 ¸ 2 F c (' ¤ Ã)(y)<br /> (F c l)(y) = °<br /> 1 ¡ ¸ 1 ¸ 2 Fc (' ¤ Ã)(y)<br /> <br /> <br /> <br /> 5<br />  Tµi liÖu tham kh¶o<br /> [1] H. Bateman and A. Erdelyi, Tables of integral transforms, Mc. Graw-Hill, New York,<br /> V.2, 1954.<br /> [2] Bramonitz and A. S. Tegun, Handbook of mathematical functions with ormulas, graphs<br /> and mathematical tables.<br /> [3] R. V. Churchill, " Fourier series and boundary value problems", New York, 1941.<br /> [4] V. A. Kakichev, On the convolution for integral transforms, Izv. AN BSSR, Ser. Fiz.<br /> Mat. 1967, N.2, 48-57.<br /> [5] V. A. Kakichev and Nguyen Xuan Thao, On the design method for the generalized<br /> integral convolution, Izv. Vuzov. Mat, 1998, N.1, 31-40 (In Russian).<br /> [6] Nguyen Xuan Thao, On the generalized convolution for Stieltjes, Hilbert, Fourier<br /> cosine and sine transforms, Ukr. Mat. J. 53 (2001), 560-567. (In Russian).<br /> [7] Nguyen Xuan Thao, V.A. Kakichev and Vu Kim Tuan, On the generalized convolution<br /> for Fourier cosine and sine transforms, East- West J. Math. 1 (1998), 85-90.<br /> [8] Nguyen Xuan Thao and Trinh Tuan, On the generalized convolution for I- transform,<br /> Acta. Mat. Vietnamica. 18 (2003), 135-145.<br /> [9] M. Saigo, S.B. Yakubovich, On the theory of convolution integral for G-transforms,<br /> Fukuoka: Univ. Sci: Reports.-1991, V. 21, N.2, 181-193<br /> [10] I. N. Sneddon, Fourier Transform, Mc. Gray Hill, New York, 1951.<br /> [11] H. M. Srivastava and Vu Kim Tuan, A new convolution theorem for the Stieltjes<br /> transform and its application to a class of singular integral equations, Arch. Math.<br /> 1995, V. 64, 144-149.<br /> [12] F. G. Tricomi, On the finite Hilbert transform, Quart. J. Math. 2 (1951), 199-211.<br /> [13] Vu Kim Tuan and Saigo M., Convolution of Hankel transform and its applications to<br /> an integral involving Bessel function of first kind, J. Math. and Math. Sci. 1995, V.18,<br /> N.2, 545--550.<br /> [14] S. B. Yakubovich, On the convolution for Kotorovich-Lebedev integral transforms and<br /> its application to integral transform, DAN. BSSR, 31 (1987), 101-103 (in Russian).<br /> [15] S. B. Yakubovich, On the construction method for construction of integral convolution,<br /> DAN. BSSR, 34 (1990), 588-591.<br /> [16] S. B Yakubovich and A. I. Mosinski, Integral equation and convolution for transforms<br /> of Kontorovich-Lebedev type, Dif. Uravnenia 29 (1993), 1272-1284.<br /> [17] S. B. Yakubovich and Nguyen Thanh Hai, Integral convolutions for H-transforms, Izv.<br /> Vuzov. Mat. 1991, N.8, 72-79.<br /> <br /> Abstracts<br /> ON THE GENERALIZED CONVOLUTION WITH A WEIGHT FUNCTION FOR THE<br /> FOURIER COSINE AND KONTOROVICH – LEBEDEV INTEGRAL TRANSFORMS<br /> The generalized convolutions with a with a wight function for the Fourier cosin and<br /> Kontorvil-Lebedev integral transforms are introduced and studied. Application for these new<br /> convolutions to solving systems of integral equations are suggested.<br /> <br /> <br /> <br /> <br /> 6<br />