Toán học Phép biến đổi tích phân kiểu tích chập suy rộng Hartley và ứng dụng

Header Page of 258 B GIO DC V O TO TRNG I HC BCH KHOA H NI HONG TH VN ANH PHẫP BIN I TCH PHN KIU TCH CHP SUY RNG HARTLEY V NG DNG Chuyờn ngnh: Toỏn gii tớch Mó s: 62460102 LUN N TIN S TON HC NGI HNG DN KHOA HC: PGS TS NGUYN XUN THO H Ni - 2016 Footer Page of 258 Header Page of 258 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca riờng tụi, di s hng dn ca PGS TS Nguyn Xuõn Tho Cỏc kt qu lun ỏn l trung thc v cha tng c cụng b cỏc cụng trỡnh ca cỏc tỏc gi khỏc Cỏn b hng dn PGS.TS Nguyn Xuõn Tho Footer Page of 258 Tỏc gi Hong Th Võn Anh Header Page of 258 LI CM N Lun ỏn c nghiờn cu v hon thnh di s hng dn ca PGS TS Nguyn Xuõn Tho, ngi luụn quan tõm, ng viờn v ch dn tỏc gi nghiờn cu khoa hc Tỏc gi xin c by t lũng bit n sõu sc v s quý mn i vi thy Tỏc gi xin c by t lũng bit n sõu sc n cỏc giỏo s, cỏc thy-cụ v cỏc ng nghip seminar Gii tớch - i s, Trng i hc Khoa hc T nhiờn-HQGHN, seminar Gii tớch, Trng i hc Bỏch khoa H Ni Nhng ý kin ca cỏc giỏo s v cỏc ng nghip tham d cỏc semina ny ó giỳp tỏc gi trng thnh hn nghiờn cu khoa hc c bit, nhng ng viờn, nhn xột quý bỏu v ý kin úng gúp sõu sc ca GS TSKH Nguyn Vn Mu, PGS TS Trn Huy H, PGS TS Nguyn Thy Thanh, PGS TS H Tin Ngon, TS Nguyn Vn Ngc, PGS TS Trnh Tuõn, TS Nguyn Thanh Hng, TS Nguyn Minh Khoa, TS Nguyn Hu Th, l nhng kinh nghim quý bỏu tỏc gi hon thnh lun ỏn mt cỏch thun li Tỏc gi xin c by t lũng bit n chõn thnh n cỏc thy cụ B mụn Toỏn c bn, Ban lónh o v cỏc thy cụ, cỏc ng nghip ca Vin Toỏn ng dng v Tin hc ó to mt mụi trng hc v nghiờn cu thun li giỳp tỏc gi honh thnh lun ỏn ny Tỏc gi xin chõn thnh cm n s quan tõm v ch dn tn tỡnh v cỏc th tc hnh chớnh ca Lónh o v cỏc anh ch cụng tỏc ti vin Sau i hc, quỏ trỡnh tỏc gi hc v nghiờn cu ti Trng i hc Bỏch khoa H Ni Tỏc gi xin by t lũng ngng m v bit n sõu sc n GS TSKH V Kim Tun, trng i hc West Georgia, Carrollton, GA 30118, USA, ngi ó luụn cú nhng ch dn, gúp ý chõn thnh v sõu sc quỏ trỡnh nghiờn cu khoa hc v hon thnh lun ỏn ca tỏc gi Tỏc gi xin c by t lũng bit n n Ban giỏm hiu, cỏc phũng ban v cỏc ng nghip ca Trng Cao ng Cụng nghip Thc phm ó khuyn khớch, ng viờn v to iu kin thun li quỏ trỡnh tỏc gi hc tp, nghiờn cu, cụng tỏc v hon thnh lun ỏn Gia ỡnh luụn l ng lc to ln i vi tỏc gi Cụng sc v s ng viờn ca i gia ỡnh l nhng úng gúp thiờng liờng ó giỏn tip giỳp tỏc gi vt Footer Page of 258 Header Page of 258 qua nhiu th thỏch hon thnh lun ỏn Tỏc gi xin c by t lũng bit n n b m, chng, hai trai v anh em hai bờn ni - ngoi Tỏc gi Footer Page of 258 Header Page of 258 MC LC LI CAM OAN LI CM N MC LC CC Kí HIU DNG TRONG LUN N M U Chng KIN THC CHUN B 1.1 Tớch chp v tớch chp suy rng 1.1.1 Mt s tớch chp ó bit 1.1.2 Tớch chp suy rng 1.1.3 Mt s nh lý quan trng 1.2 Bt ng thc tớch chp 1.2.1 Cỏc bt ng thc tớch phõn 1.2.2 Bt ng thc tớch chp 1.3 Mt s hm c bit 11 24 24 24 27 29 29 30 31 33 Chng TCH CHP SUY RNG HARTLEY 2.1 Tớch chp suy rng Hartley-Fourier sine 2.1.1 nh ngha v cỏc tớnh cht 2.1.2 Tớch chp suy rng liờn quan n phộp bin i Hartley 2.1.3 ng dng 2.2 Tớch chp suy rng Hartley-Fourier cosine 2.2.1 nh ngha v cỏc tớnh cht 2.2.2 Phng trỡnh v h phng trỡnh Toeplitz-Hankel 37 37 37 45 48 58 58 62 Chng BT NG THC TCH CHP SUY RNG V NG DNG 3.1 Bt ng thc kiu Hausdorff - Young 3.2 Bt ng thc tớch chp suy rng Hartley-Fourier cosine 3.3 Bt ng thc tớch chp suy rng Hartley-Fourier sine 3.4 ng dng 72 72 75 85 88 Footer Page of 258 khụng gian Header Page of 258 3.4.1 3.4.2 3.4.3 3.4.4 Phng trỡnh tớch phõn kiu Toeplitz-Hankel Phng trỡnh vi phõn Bi toỏn Dirichlet trờn na mt phng Bi toỏn Cauchy cho phng trỡnh truyn nhit 88 89 91 92 Chng PHẫP BIN I TCH PHN KIU TCH CHP SUY RNG HARTLEY 4.1 Cỏc tớnh cht toỏn t 4.1.1 nh lý kiu Watson 4.1.2 nh lý kiu Plancherel 4.1.3 Tớnh b chn ca toỏn t vi-tớch phõn 4.1.4 Vớ d 4.2 ng dng 4.2.1 Phng trỡnh vi-tớch phõn 4.2.2 Phng trỡnh parabolic tuyn tớnh 4.2.3 H phng trỡnh vi-tớch phõn KT LUN KIN NGH HNG NGHIấN CU TIP THEO DANH MC CC CễNG TRèNH CễNG B CA LUN N TI LIU THAM KHO Footer Page of 258 95 96 96 100 103 105 108 108 112 112 117 118 119 120 Header Page of 258 CC Kí HIU DNG TRONG LUN N a Kớ hiu tớch chp, tớch chp suy rng v phộp bin i tớch phõn Cỏc tớch chp, tớch chp suy rng (ã ã) l tớch chp i vi phộp bin i Fourier (ã ã) l tớch chp i vi phộp bin i Laplace (ã ã) l tớch chp i vi phộp bin i Fourier cosine (ã ã) l tớch chp i vi phộp bin i Fourier sine (ã ã), (ã ã), (ã ã) l tớch chp, cỏc tớch chp suy rng i vi cỏc phộp F L Fc Fs H H11 H12 bin i tớch Hartley (ã ã) l tớch chp suy rng i vi cỏc phộp bin i Fourier sine v Fs Fc Fourier cosine (ã ã) l tớch chp suy rng vi hm trng (y) = sin y i vi cỏc phộp Fc Fc bin i Fourier sine v Fourier cosine (ã ã) l tớch chp suy rng i vi cỏc phộp bin i Fourier cosine v Fs Fs sine (ã ã) l tớch chp suy rng i vi cỏc phộp bin i Hartley, Fourier HF (ã ã) l tớch chp suy rng i vi cỏc phộp bin i Hartley, Fourier sine (ã ã) l tớch chp suy rng i vi cỏc phộp bin i Hartley, Fourier cosine Cỏc phộp bin i tớch phõn Phộp bin i cosine, phộp bin i sine (Tc f )(y) := (Ts f )(y) := f (x) cos(xy) dx, y R, f (x) sin(xy) dx, y R Footer Page of 258 Header Page of 258 Phộp bin i Hartley (H1 f )(y) = f (x) cas(xy)dx, (H2 f )(y) = f (x) cas(xy)dx, ú cas u := cos u + sin u l nhõn ca phộp bin i tớch phõn Hartley Phộp bin i Fourier (F f )(x) = eixy f (y)dy, y R Phộp bin i Fourier ngc (F g)(x) = eixy g(y)dy, y R Phộp bin i Fourier cosine (Fc f )(y) = f (x) cos(xy) dx, y R+ Phộp bin i Fourier cosine ngc (Fc1 g)(x) = g(y) cos(xy) dy, y R+ Phộp bin i Fourier sine (Fs f )(y) = f (x) sin(xy) dx, Footer Page of 258 y R+ Header Page of 258 Phộp bin i Fourier sine ngc (Fs1 g)(x) = g(y) sin(xy) dy, y R+ Th , Th1 l phộp bin i tớch phõn kiu tớch chp suy rng Hartley-Fourier cosine v phộp bin i ngc ca nú d2 (h f )(x), dx2 (Th f )(x) := d2 (h g)(x) dx2 Tk , Tk1 l phộp bin i tớch phõn kiu tớch chp suy rng Hartley-Fourier sine v phộp bin i ngc ca nú f (x) = Th1 g (x) := d2 (k f )(x), dx2 (Tk f )(x) := f (x) = Tk1 g (x) := d2 (k g)(x) dx2 b Cỏc khụng gian hm R+ = {x R, x > 0} Lp (R+ ), p < l hp cỏc hm s f (x) xỏc nh trờn R+ cho |f (x)|p dx < f Lp (R+ ) l chun ca hm f khụng gian Lp (R+ ), xỏc nh bi f Lp (R+ ) p |f (x)| dx := p f Lp (R) l chun ca hm f khụng gian Lp (R), xỏc nh bi f Lp (R) p |f (x)| dx := Footer Page of 258 p Header Page 10 of 258 L (R+ ) l hp cỏc hm s f (x) xỏc nh trờn R+ cho sup |f (x)| < xR+ f L (R) l chun ca hm f khụng gian L (R), xỏc nh bi f L (R) := sup |f (x)| R Lp (R+ , ), cho p < l hp cỏc hm s f (x) xỏc nh trờn R+ |f (x)|p (x)dx < , ú l mt hm trng dng f Lp (R+ ,) l chun ca hm f khụng gian Lp (R+ , ), xỏc nh bi f Lp (R+ ,) p |f (x)| (x)dx := p (R) l khụng gian ba tham s, xỏc nh bi L,, p L,, (R) := Lp R, |x| e|x| p f L,, (R) p , R, < < 1, > (R), xỏc nh bi l chun ca hm f khụng gian L,, p f L,, (R) p 1/p := |f (x)|p |x| e|x| dx c Cỏc hm c bit Erf(z), Erfi(z) tng ng l hm sai s (error function) v hm sai s o (imaginary error function) (z) l hm Gamma Gm,n p,q (ã) l hm Meijer G J (x), Y (x) tng ng l hm Bessel loi mt, hm Bessel loi hai I (z), K (z) l cỏc hm Bessel suy bin pF q(a; b; z) l hm siờu bi suy rng 10 Footer Page 10 of 258 Header Page 111 of 258 Vy phng trỡnh (4.40) tr thnh (H1 f )(y) = (Fc l)(y) (H1 g) = (H1 g)(y) H1 (l g)(y) Suy nghim nht ca phng trỡnh bi toỏn (4.33) cú dng (4.36) Vỡ g L1 (R), l L1 (R+ ) v theo tớnh cht ca tớch chp suy rng HartleyFourier cosine, suy f L1 (R) nh lý c chng minh Nhn xột 4.2.1 T bi toỏn (4.33), ta xột trng hp riờng k = 1, ú n k=1 d2 d2 + (2k 1) = dx dx Do ú, bi toỏn (4.33) tr thnh f (x) + (Th f )(x) = g(x), x R, (4.44) vi iu kin nh sau f (0) = 0, lim f (x) = lim f (x) = 0, x x ú f l n hm, g L1 (R), h L1 (R+ ) l nhng hm ó bit Ta d dng chng minh c nghim ca bi toỏn ny bi nh lý sau õy nh lý 4.2.2 Gi s hm h1 L1 (R+ ) tha iu kin 1+ y sech y + (Fc h1 )(y) = 0, y R+ , 2 (4.45) l L1 (R+ ) l hm c xỏc nh nh sau (Fc l)(x) = 1+ sech y y + (Fc h1 )(y) y + (F h )(y) sech y c 2 (4.46) Khi ú, phng trỡnh vi-tớch phõn (4.44) cú nghim nht L1 (R) xỏc nh bi cụng thc f (x) = g(x) (l g)(x), x R 111 Footer Page 111 of 258 (4.47) Header Page 112 of 258 4.2.2 Phng trỡnh parabolic tuyn tớnh Xột phng trỡnh parabolic sau õy u(x, t) u(x, t) = Th (u)(x, t) t x2 (4.48) ú, u(x, t) l hm cha bit, ta chn hm nhõn h(y) cho (Fc h)(y) = , + y2 ú tha iu kin (4.3) p dng phộp bin i Hartley i vi x cho c hai v ca phng trỡnh (4.48), v t (H1 u)(y, t) = U (y, t) ta nhn c phng trỡnh vi phõn sau d U (y, t) = y U (y, t) (1 + y )(Fc h)(y)U (y, t), dt tng ng vi d U (y, t) = (1 + y )U (y, t) dt Khi ú, nghim ca phng trỡnh ny cú dng U (y, t) = e(1+y )C(y)t (4.49) Chn C(y) v ỏp dng phộp bin i Hartley ngc, ta nhn c nghim ca phng trỡnh (4.48) nh sau u(x, t) = x2 2et 4t x + Erfi t t , (4.50) õy Erfi(t) l hm sai s o 4.2.3 H phng trỡnh vi-tớch phõn Xột h hai phng trỡnh vi-tớch phõn i vi phộp bin i tớch phõn kiu tớch chp suy rng Hartley-Fourier cosine cú dng f (x) + (Th g)(x) = p(x), g(x) + (Th f )(x) = q(x), 112 Footer Page 112 of 258 (4.51) Header Page 113 of 258 vi cỏc iu kin ban u f (0) = 0, g (0) = 0, lim f (x) = lim f (x) = = lim g(x) = lim g (x) x x x (4.52) x Trong ú, cỏc toỏn t Th , Th xỏc nh bi d2 1 (Th g)(x) := dx [g(x + u) + g(x u)]h(u)du , d2 Th f (x) := dx [f (x + u) + f (x u)]h(u)du , v cỏc hm h, h L1 (R+ ) xỏc nh nh sau h(x) = h1 ( ) sech (x), h(x) = h1 ( ) sech (x), Fc Fc (4.53) vi f, g L1 (R) l cỏc n hm; h, h1 , h, h1 , p, q l cỏc hm ó bit, v cỏc hm h1 , h1 L1 (R+ ); p, q L1 (R) nh lý sau õy cho phộp ta xỏc nh c iu kin cú nghim v cụng thc nghim ca h phng trỡnh vi-tớch phõn (4.51) nh lý 4.2.3 Gi s rng iu kin sau c tha y (1 + y )2 sech2 |(Fc h1 )(y)|2 = 0, y R, 2 (4.54) v gi s hm l L1 (R+ ) xỏc nh nh sau y (1 + y )2 sech2 |(Fc h1 )(y)|2 (Fc l)(y) = y 2 (1 + y ) sech |(Fc h1 )(y)|2 2 (4.55) Khi ú, h phng trỡnh vi-tớch phõn (4.51) vi iu kin ban u (4.52) cú nghim nht khụng gian L1 (R) ì L1 (R) xỏc nh bi f (x) =p(x) + (l p)(x) (h1 sech3 ) q (x) l (h1 sech3 ) q Fc 113 Footer Page 113 of 258 Fc (x) Header Page 114 of 258 g(x) =q(x) + (l q)(x) (h1 sech3 ) p (x) l (h1 sech3 ) p 2 Fc Fc (x) (4.56) Chng minh p dng phộp bin i Hartley H1 cho c hai v ca phng trỡnh th nht h (4.44) v ng thc nhõn t húa (2.46), ta cú (H1 p)(y) = (H1 f )(y) + (1 + y )H1 (h g)(y) 2 = (H1 f )(y) + (1 + y )(Fc h)(y) ã (H1 g)(y) Theo cụng thc (4.53), (1.11) v (1.9.4) [10], ta nhn c (H1 p)(y) = (H1 f )(y) + (1 + y )[(Fc h1 )(y) ã Fc (sech )(y)](H1 g)(y) = (H1 f )(y) + (1 + y )(Fc h1 )(y) ã Fc (sech )(y) ã (H1 g)(y) y = (H1 f )(y) + (1 + y ) sech (Fc h1 )(y) ã (H1 g)(y) 2 = (H1 f )(y) + 2Fc (sech )(y)(Fc h1 )(y) ã (H1 g)(y), tng ng vi (H1 p)(y) = (H1 f )(y) + 2Fc (h1 sech3 )(y) ã (H1 g)(y) Fc (4.57) Bng k thut tng t ta cú (H1 q)(y) = (H1 g)(y) + (1 + y )H1 (h f )(y), suy (H1 q)(y) = (H1 g)(y) + 2Fc (h1 sech3 )(y) ã (H1 f )(y) Fc (4.58) T cỏc cụng thc (4.57) v (4.58) suy h (4.51) vit c di dng (H1 f )(y) + 2Fc (h1 sech3 )(y) ã (H1 g)(y) = (H1 p)(y), Fc 2Fc (h1 sech3 )(y) ã (H1 f )(y) + (H1 g)(y) = (H1 q)(y) Fc 114 Footer Page 114 of 258 (4.59) Header Page 115 of 258 Xột h (4.59) ta nhn c = 4Fc (h1 sech3 ) (h1 sech3 ) (y) Fc Fc Fc y = (1 + y )2 sech2 |(Fc h1 )(y)|2 2 (4.60) T iu kin (4.54), theo nh lớ Wiener-Lộvy 1.1.1 tn ti hm l L1 (R+ ) cho cụng thc (4.55) tha Do ú, ta nhn c 4Fc (h1 sech3 ) (h1 sech3 ) (y) Fc Fc Fc =1+ 4Fc (h1 sech3 ) (h1 sech3 ) (y) Fc Fc Fc y (1 + y )2 sech2 |(Fc h1 )(y)|2 =1+ = + (Fc l)(y) y 2 2 (1 + y ) sech |(Fc h1 )(y)| 2 Khi ú, gii h (4.59) i vi (H1 f )(y) v (H1 g)(y), ta cú = (H1 p)(y) 2H1 (h1 sech3 ) q) (y), Fc = (H1 q)(y) 2H1 (h1 sech3 ) p) (y) Fc T ú, nhn c h sau (H1 f )(x) =(H1 p)(x) + H1 (l p)(x) 2H1 (h1 sech3 ) q (x) Fc 2H1 l (h1 sech3 ) q Fc 2 (x) (H1 g)(x) =(H1 q)(x) + H1 (l q)(x) 2H1 (h1 sech3 ) p (x) Fc 2H1 l (h1 sech3 ) p Fc 2 (x) Nhn thy rng, cỏc biu thc trờn l ỳng vi mi x R, iu ny tng ng vi nghim cn tỡm ca h phng trỡnh vi-tớch (4.51) l cụng thc (4.56) Do cỏc hm p, q thuc khụng gian L1 (R), cỏc hm h, h1 , h, h1 , l L1 (R+ ), nờn theo nh lý 2.2.1, suy f L1 (R), g L1 (R), v chỳng tha iu kin (4.54) nh lý c chng minh 115 Footer Page 115 of 258 Header Page 116 of 258 Kt lun chng Trong chng ny ó nhn c mt s kt qu sau: Xõy dng hai phộp bin i tớch phõn kiu tớch chp suy rng HartleyFourier cosine, Hartley-Fourier sine v nhn c cỏc toỏn t ngc tng ng Nhn c cỏc nh lý kiu Watson, kiu Plancherel L2 (R) Chng minh c tớnh b chn ca cỏc toỏn t tớch phõn ny khụng gian Lp (R), p Xõy dng c cỏc vớ d c th minh cho s tn ti ca cỏc toỏn t tớch phõn kiu tớch chp suy rng ó nghiờn cu, lm rừ hn s tn ti ca cỏc phộp bin i tớch phõn trờn ng dng gii phng trỡnh v h phng trỡnh vi-tớch phõn, nhn c cụng thc biu din nghim ca lp phng trỡnh o hm riờng parabolic 116 Footer Page 116 of 258 Header Page 117 of 258 KT LUN Cỏc kt qu chớnh ca lun ỏn ó t c: Xõy dng c cỏc tớch chp suy rng Hartley mi nh: Hartley-Fourier sine, Hartley-Fourier cosine, Hartley-Fourier, Hartley H1 v H2 Nhn c cỏc ng thc nhõn t húa, ng thc Parseval v nh lý kiu Titchmarch, nh lý kiu Wiener-Levy Nhn c cỏc bt ng thc tớch chp suy rng kiu Saitoh, kiu Saitoh ngc, kiu Young v kiu Hausdorff-Young ca cỏc tớch chp suy rng mi xõy dng p dng cỏc bt ng thc thu c a cỏc ỏnh giỏ nghim ca phng trỡnh tớch phõn kiu Toeplitz-Hankel v mt s bi toỏn Toỏn-Lý Xõy dng c hai phộp bin i tớch phõn kiu tớch chp suy rng Hartley-Fourier cosine Th , Hartley-Fourier sine Tk v nhn c cỏc toỏn t ngc tng ng Th1 , Tk1 Nhn c cỏc nh lý kiu Watson, nh lý kiu Plancherel L2 (R), tớnh b chn khụng gian Lp (R) vi p ng dng cỏc kt qu nhn c gii mt lp phng trỡnh v h phng trỡnh tớch phõn nhõn Toeplitz-Hankel, phng trỡnh vi phõn, phng trỡnh v h phng trỡnh vi-tớch phõn, phng trỡnh o hm riờng parabolic mt chiu 117 Footer Page 117 of 258 Header Page 118 of 258 KIN NGH HNG NGHIấN CU TIP THEO Mt s cn nghiờn cu tip theo nh sau: Nghiờn cu cỏc tớch chp, tớch chp suy rng i vi phộp bin i tớch phõn Hartley trờn thang thi gian v ng dng vo cỏc bi toỏn toỏn lý M rng cỏc kt qu nghiờn cu lun ỏn cỏc khụng gian n chiu (n 2) Nghiờn cu cỏc ng dng ca lun ỏn vo vic gii mt s bi toỏn thc tin 118 Footer Page 118 of 258 Header Page 119 of 258 DANH MC CC CễNG TRèNH CễNG B CA LUN N Thao N.X., Tuan V.K, and Anh H.T.V (2014), On the Toeplitz plus Hankel integral equation II, Integral Transforms and Special Functions, Vol 25, No 1, pp 75-84, (ISI) Thao N.X., and Anh H.T.V (2014), On the Hartley - Fourier sine generalized convolution, Mathematical Methods in the Applied Sciences, Vol 37, Issue 15, pp 2308-2319, (ISI) Anh H.T.V., and Thao N.X (2015), Hartley-Fourier cosine generalized convolution inequalities, Mathematical Inequalities and Applications, Vol 18, No 4, pp 1393-1408, (ISI) Thao N.X., and Anh H.T.V (2015), Hartley-Fourier sine generalized convolution inequalities, K yu hi ngh quc t v ng dng toỏn hc, nh xut bn Thụng tin v Truyn thụng, pp 120-131 119 Footer Page 119 of 258 Header Page 120 of 258 TI LIU THAM KHO [1] ng ỡnh ng (2009), Bin i tớch phõn, Nh xut bn Giỏo dc, H Ni [2] Nguyn Thy Thanh (2007), C s lý thuyt hm bin phc, Nh xut bn i hc Quc gia H Ni [3] Nguyn Vn Khuờ, Lờ Mu Hi (2010), Giỏo trỡnh gii tớch hm, Nh xut bn i hc S phm H Ni [4] Nguyn Vn Mu (2006), Lý thuyt toỏn t v phng trỡnh tớch phõn k d, Nh xut bn i hc Quc gia H Ni [5] Phan Quc Khỏnh (2000), Toỏn chuyờn , Nh xut bn i hc Quc gia Thnh ph H Chớ Minh Ting Anh [6] Adams R.A., and Fournier J.J.F (2003), Sobolev Spaces, 2nd ed., Academic Press, New York, Amsterdam, Elsevier Science [7] Al-Musallam F., and Tuan V.K (2000), Integral transforms related to a generalized convolution, Results in Mathematics, Vol 38, No 3, pp 197208 [8] Al-Musallam F., and Tuan V.K (2000), A class of convolution transforms, Fractional Calculus and Applied Analysis, Vol 3, Issue 3, pp 303-314 [9] Anh P.K., Tuan N.M., and Tuan P.D (2013), The finite Hartley new convolutions and solvability of the integral equations with Toeplitz plus Hankel kernels, Journal of Mathematical Analysis and Applications, Vol 397, Issue 2, pp 537549 [10] Bateman H., and Erdelyi A (1954), Table of Integral Transforms, New York-Toronto-London, Vol I McGraw-Hill Book Company, Inc 120 Footer Page 120 of 258 Header Page 121 of 258 [11] Băottcher A., and Silbermann B (2009), Analysis of Toeplitz Operators: Second Edition, Springer-Verlag, Berlin-Heidelberg-New York [12] Bracewell R.N (1986), The Hartley transform, Oxford University Press, Clarendon Press, New York [13] Britvina L.E (2005), A class of integral transforms related to the Fourier cosine convolution, Integral Transforms and Special Functions, Vol 16, Issue 5-6, pp 379-389 [14] Duoadikoetxea J (2001), Fourier Analysis, AMS Providence, Rhode Island [15] Bouzeffour F (2014), The generalized Hartley transform, Integral Transforms and Special Functions, Vol 25, Issue 3, pp 230-239 [16] Giang B.T., Mau N.V., and Tuan N.M (2009), Operational properties of two integral transforms of Fourier type and their convolutions, Integral Equations and Operator Theory, Vol 65 Issue 3, pp 363-386 [17] Giang B T., and Tuan N M (2010), Generalized convolutions and the integral equations of the convolution type, Complex Variables and Elliptic Equations, Vol 55, No 4, pp 331345 [18] Gradshteyn, Ryzhik (2007), Tables of integrals, series, and products, 7ed., Academic Press [19] Hai N.T., Yakubovich S B., and Wimp J (1992), Multidimensional Watson transform International Journal of Mathematics and Statistics, Vol 1, No 1, pp 105-119 [20] Hong N.T (2010), Fourier cosine convolution inequalities and applications, Integral Transforms Special Functions, Vol 21, Issue 10, pp 755-763 [21] Paraskevas I., Barbarosou M., and Chilton E (2015), Hartley transform and the use of the Whitened Hartley spectrum as a tool for phase spectral processing, The Journal of Engineering, Accepted on 9th February [22] Kakichev V.A (1967), On the convolution for integral transforms, Izvestiya Vysshikh Uchebnykh Zavedenii Matematika, Vol 2, pp 53-62 (In Russian) 121 Footer Page 121 of 258 Header Page 122 of 258 [23] Kakichev V.A., and Thao N.X (1998), On the design method for the generalized integral convolutions, Izvestiya Vysshikh Uchebnykh Zavedenii Matematika, Vol 1, pp 31-40 (In Russian) [24] Kakichev V.A., Thao N.X., and Tuan V.K (1998), On the generalized convolutions for Fourier cosine and sine transforms, East- West Journal of Mathematics, Vol 1, No 1, pp 85-90 [25] Luchko Y (2008), Integral transforms of the Mellin convolution type and their generating operators, Integral Transforms and Special Functions, Vol 19, Issue 11, pp 809-851 [26] Nhan N.D.V., and Duc D.T (2008), Reverse weighted Lp norm inequalities and their applications, Journal of Mathematical Inequalities, Vol 2, No 1, pp 57-73 [27] Nhan N.D.V., Duc D.T., and Tuan V.K (2009), On some reverse weighted Lp (R+ ) norm inequalities in convolution and their applications, Mathematical Inequalities and Applications, Vol 12, No 1, pp 67-80 [28] Nikiforov F., and Uvarov B (1988), Special Functions of Mathematical Physics: A Unified Introduction with Applications Birkhauser Verlag Basel [29] Olejniczak K.J (2000), The Hartley transform, The Transform and Applications Handbook, edited by A D Poularikas, 2nd Edition, The Electrical Engineering Handbook Series, CRC Press with IEEE Press, Florida, pp 341-401 [30] Paley R.E.A.C., and Wiener N (1934), Fourier Transforms in the Complex Domain AMS, New York [31] Poularikas A.D (2010), Transforms and Applications Handbook, CRC Press, Taylor and Francis Group [32] Przeworska-Rolewicz D., and Rolewicz S (1968), Equations in Linear Spaces, PWN-Polish Scientific Pub., Warszawa [33] Sneddon I.N (1951), Fourier Transforms, McGray-Hill, New York 122 Footer Page 122 of 258 Header Page 123 of 258 [34] Sneddon I.N (1972), The Use of Integral Transforms, Mc Gray-Hill NewYork [35] Stein E.M., and Weiss G (1971), Introduction to Fourier Analysis on Euclidean Space, Princeton Univ Press [36] Saitoh S (2000), Weighted Lp -norm inequalities in convolution, Survey on Classical Inequalities, Kluwer Academic Publishers, Amsterdam, pp 225234 [37] Saitoh S., Tuan V.K., and Yamamoto M (2003), Convolution inequalities and applications, Journal of Inequalities in Pure and Applied Mathematics, Vol 4, Issue 3, pp 1-8 [38] Saitoh S., Tuan V.K., and Yamamoto M (2000), Reverse weighted Lp norm inequalities in convolutions and stability in inverse problems, Journal of Inequalities in Pure and Applied Mathematics, Vol.1, Issue 1, Article [39] Saitoh S., Tuan V.K., and Yamamoto M (2002), Reverse convolution inequalities and applications to inverse heat source problems, Journal of Inequalities in Pure and Applied Mathematics, Vol 3, Issue 5, pp 1-11 [40] Thao N.X., and Khoa N.M (2006), On the generalized convolution with a weight-function for the Fourier sine and cosine transforms, Integral Transforms and Special Function, Vol 17, No 9, pp 673-685 [41] Thao N.X., Tuan V.K., and Hong N.T (2007), Integral transforms of Fourier cosine and sine generalized convolution type, International Journal of Mathematics and Mathematical Sciences, pp 1-11 [42] Hong N.T., Tuan T., and Thao N.X (2013), On the Fourier cosineKontorovich-Lebedev generalized convolution transforms, Applications of Mathematics, Vol 58, Issue 4, pp 473-486 [43] Thao N.X., Tuan V.K., and Hong N.T (2008), Generalized convolution transforms and Toeplitz plus Hankel integral equations, Fractional Calculus and Applied Analysis, Vol 11, No 2, pp 153-174 123 Footer Page 123 of 258 Header Page 124 of 258 [44] Thao N.X., and Hong N.T (2008), Integral transforms related to the Fourier sine convolution with a weight function, Vietnam Journal of Mathematics, Vol 6, No 1, pp 83-101 [45] Thao N.X., Tuan V.K., and Hong N.T (2011), Toeplitz plus Hankel integral equations, Integral Transforms and Special Functions, Vol 22, Issue 10, pp 723-737 [46] Titchmarch E.C (1986), Introduction to the Theory of Fourier integrals, 3rd Ed, Chelsea publishing Co., NewYork [47] Tsitsiklis J N., and Levy B.C (1981), Integral equations and resolvents of Toeplitz plus Hankel kernels, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology Series/Report No.: LIDSP 1170 [48] Vilenkin Y.Ya (1958), Matrix elements of midecomsale unitary representations for motions group of the Lobachevskiis space and generalized MehlerFox transforms, Doklady Akademii Nauk SSSR, Vol 118, No 2, pp 219222 (In Russian) [49] Tuan V.K., and Yakubovich S.B (1992), A criterion for a two-sided integral transform to be unitary Ukrainian Mathematical Journal, Vol 44, Issue 5, pp 630-632 [50] Tuan V.K (1999), Integral transforms of Fourier cosine convolution type, Journal of Mathematical Analysis and Applications, Vol 229, Issue 2, pp 519-529 [51] Xiao-Hua L (1990), On the inverse of Hă older inequality, Math Practice and Theory, Vol 1, pp 84-88 [52] Yakubovich S.B (1990), On the construction method for construction of integral convolution, Doklady Akademii Nauk SSSR, Vol 34, No 7, pp 588-591 [53] Yakubovich S.B., and Luchko Yu.F (1991), The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Academic Publishers 124 Footer Page 124 of 258 Header Page 125 of 258 [54] Yakubovich S.B., and Mosinski A.I (1993), Integral-equation and convolutions for transform of Kontorovich-Lebedev type, Differential Equations (Differentsialnye Uravneniya), Vol 29, No 7, pp 1107-1118 (In Russian) [55] Yakubovich S.B (2003), Integral transforms of the Kontorovich-Lebedev convolution type, Collectanea Mathematica, Vol 54, No 2, pp 99-110 [56] Yakubovich S.B (2014), On the half-Hartley transform, its iteration and compositions with Fourier transforms, J Integral Equations Applications, Vol 26, No 4, pp 581-608 [57] Yakubovich S.B (2014), The Plancherel, Titchmarsh and Convolution theorems for the half-Hartley transform, Integral Transforms and Special Functions, Vol 25, Issue 10, pp 836-848 [58] Yakubovich S.B (2004), New inversion, Convolution and Titchmarshs theorems for the half-Hartley transform, arXiv preprint arXiv:1401.3143 125 Footer Page 125 of 258 ... 33 Chng TCH CHP SUY RNG HARTLEY 2.1 Tớch chp suy rng Hartley- Fourier sine 2.1.1 nh ngha v cỏc tớnh cht 2.1.2 Tớch chp suy rng liờn quan n phộp bin i Hartley 2.1.3 ng... cỏc hm c bit Chng 2, xõy dng cỏc tớch chp suy rng Hartley mi l tớch chp suy rng Hartley- Fourier cosine, Hartley- Fourier sine, Hartley- Fourier, Hartley H1 v H2 Chng minh cỏc ng thc nhõn t húa,... tớch phõn kiu tớch chp suy rng Hartley v ng dng" Mc ớch, i tng v phm vi nghiờn cu Mc ớch: - Xõy dng mt s tớch chp suy rng Hartley Nghiờn cu cỏc tớnh cht ca cỏc tớch chp suy rng ny v ng dng gii