On Approximation of Conjugate of Signals (Functions) Belonging to the Generalized Weighted. of Conjugate Series of Fourier Series
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1 I Joual of Mah Aalyi, Vol 6,, o 35, O Aoximaio of ojugae of Sigal (Fucio Belogig o he Geealized Weighed ( x W L, (, ( ³ -la y Poduc Summailiy Mea of ojugae Seie of Fouie Seie Vihu Naaya Miha, Huzoo H Kha, Kejal Khai ad Lahmi Naaya Miha 3 Deame of Alied Mahemaic & Humaiie, SV Naioal Iiue of Techology, Ichchhaah Mahadev Road, Sua, Sua (Gujaa Idia vihu_aayamiha@yahoocoi, ejal99@gmailcom Deame of Mahemaic, Aligah Mulim Uiveiy, Aligah - (UP, Idia, huzooha@yahoocom 3 D Ram Maoha Lohia Avadh Uiveiy, Hawai Pai Allahaad Road, Faizaad, Faizaad 4 (Ua Padeh, Idia lahmiaayamiha4@gmailcom, l miha@yahoocoi Aac I hi ae, a ow heoem Nigam ad Shama [8] dealig wih he degee of aoximaio of cojugae of a fucio elogig o Lix ( -cla y ( E, q(, oduc ummailiy mea of cojugae eie of Fouie eie ha ee geealized W L, x (, ( ³,( > - cla, whee ξ( i o-egaive ad fo he Weighed ( iceaig fucio of, y E ² (Poduc Tafom of ojugae Seie of Fouie q eie Mahemaic Sujec laificaio: Pimay 4B5, 4B8 Keywod: Degee of aoximaio, weighed ( W L, x (, ( ³,( > - cla of fucio, ( E, q mea, (, mea, ( E, q (, oduc mea, cojugae Fouie eie, Leegue iegal
2 74 Vihu Naaya Miha e al Ioducio Kha [5, 6] ha udied he degee of aoximaio of a fucio elogig o Li ( a, ad W ( L, x( clae y Nlud ad geealized Nlud mea Woig i he ame diecio, Mial, Rhoade ad Miha [6], Mial ad Miha [4] ad Miha [8, 9] have udied he degee of aoximaio of a fucio elogig o W ( L, x( cla y liea oeao Theeafe, Nigam ad Shama [8] dicued he degee of aoximaio of cojugae of a fucio elogig o Li ( x (, cla y ( E, q (, oduc ummailiy mea cely, Rhoade e al [] have deemied he degee of aoximaio of a fucio elogig o Lia cla y Haudoff mea Bu ohig eem o have ee doe o fa o oai he degee of aoximaio of cojugae of igal elogig o he geealized weighed W ( L, x( cla y ( E, q (, oduc ummailiy mea ( W L, x (, ( ³ - la i geealizaio of Lia, Li ( a, ad Li ( x (, clae Theefoe, i ee ae, a heoem o degee of aoximaio of cojugae of igal elogig o he geealized weighed W ( L, x (, ³ cla y ( E, q (, oduc ummailiy mea of cojugae eie of Fouie eie ha ee oved Le f ( x e a - eiodic fucio ad Leegue iegale The Fouie eie of f ( x i give y a f ( x : å ( a co x i x ( = h wih aial um ( f ; x The cojugae eie of Fouie eie ( i give y Le å ( a i x- co x ( = å u e a give ifiie eie wih equece of i = ( E, q afom i defied a he q i y E If h aial um{ } The h aial um of ( E, q ummailiy ad we deoe q - E = å ç q, a (3 ( he he ifiie eie If q = è å u i ummale (, = E q o a defiie ume [4] = = å, a (4 =
3 Aoximaio of cojugae of igal 75 he he ifiie eie å u i ummale o he defiie ume y (, = mehod The ( E, q afom of he (, afom defie ( E, q (, oduc afom ad deoe i y E q Thi if q - E = q, a å ç (5 ( q = è he he ifiie eie ummale ( E, q (, o a defiie ume A fucio f ( x Î Li a, if å u i aid o e ummale y (, (, = a ( E q mehod o f ( x - f ( x =O fo < a, > (6 ad f ( x Î Li ( a,, fo x, if ( a ( w ( ; f = f ( x - f ( x dx =O, < a, ³, > (7 Give a oiive iceaig fucio x (, f ( x Î Li( x (, if ( ( w ( ; f = f ( x - f ( x dx =O x (, ³, > (8 Give oiive iceaig fucio x (, a iege ³, f Î W ( L, x (, ([5], if { } w ( ; f = { f ( x - f ( x } i x dx =O ( x (, ( ³, > Fo ou coveiece o evaluae I wihou eo, we edefie he weighed cla a follow w ( ; f = ( ( ( i ( x f x - f x dx =O ( x (, ³, > [,7] (9 If =, he he weighed cla W ( L, x ( coicide wih he cla Li ( x (,, we oeve ha = x ( = (, x ( ¾¾¾ ( x (, ¾¾¾ a ( a, ¾¾¾ a < a, ³, > W L Li Li Li fo ( L - om of a fucio i defied y ( f = f ( x dx, ³ ( A igal (fucio f i aoximaed y igoomeic olyomial of ode ad he degee of aoximaio E ( f i give y [] /
4 76 Vihu Naaya Miha e al E ( f = mi f ( x - ( f ; x ( i em of, whee ( f ; x i a igoomeic olyomial of degee Thi mehod of aoximaio i called Tigoomeic Fouie Aoximaio (TFA [6] We ue he followig oaio houghou hi ae co y ( = f ( x - f ( x- ad ² ( v - G ( = q å ( q ç å v i = è = ë Peviou eul Nigam ad Shama [8] have oved a heoem o he degee of aoximaio of ² f ( x, cojugae o a eiodic fucio f ( x wih eiod ad elogig o he cla Li ( x (, ( ³, y ( E, q (, oduc ummailiy mea of cojugae eie of Fouie eie They have oved he followig heoem Theoem [8] If ² f ( x, cojugae o a - eiodic fucio f ( x elogig o Li ( x (, cla, he i degee of aoximaio y ( E, q (, oduc ummailiy mea of cojugae eie of Fouie eie i give y ² q E - f =O ( ç è x çè ( ovided x ( aifie he followig codiio: y ( d =O ç x ç ç ( è ( è è -d y ( d ad ç d (( ç =O ç (3 è x ( è whee d i a aiay ume uch ha ( -d - >, =,, codiio ( ad (3 hold uifomly i x ad E ² i ( E, q (, mea of he eie ( ma The oof oceed y eimaig ² q E - f, which i eeeed i em of a iegal The domai of iegaio i divided io wo a fom, ad, ë ë feig o ecod iegal a I, ad uig Hlde iequaliy, he auho [8] oai q
5 Aoximaio of cojugae of igal 77 ì -d ü ì ( x( G ( y I ï d í ï ý ï í dï -d ý ç x( è ï ç î ïþ è ïî ïþ ì ² ü ( G ( x d =O (( ï í dï ý - d ç è ïî ïþ The auho he mae he uiuio y= o oai ( y x d ç dy =O (( ç d- y y ç ë è I he ex e x ( y ² ü i emoved fom he iegad y elacig i wih O ç x ç è è while ( x i ow deceaig y fucio Theefoe, hi e i ivalid 3 Mai Theoem I i well ow ha he heoy of aoximaio ie TFA, which oigiaed fom a heoem of Weiea, ha ecome a exciig iediciliay field of udy fo he la 3 yea Thee aoximaio have aumed imoa ew dimeio due o hei wide alicaio i igal aalyi [9], i geeal ad i Digial Sigal Poceig [3] i aicula, i view of he claical Shao amlig heoem Thi ha moivaed Mial Rhoade ([ 3] ad Mial e al ([6, 7] o oai may eul o TFA uig ummailiy mehod wihou ow of he maix I hi ae, we ove he followig heoem 3 Theoem ² f x, cojugae o a - eiodic fucio f elogig o he geealized If ( weighed W( L, ( x i a iceaig fucio, ( x -la ( ³, he i degee of aoximaio y ( E, q (, oduc ummailiy mea of cojugae eie of Fouie eie i give y ² q ( E f - =O x ç è çè (3 ovided x ( aifie he followig codiio: y( ç i d =O ç ç x ( ç è è (3 è
6 78 Vihu Naaya Miha e al ad -d y( d ç d =O (( ç ç x ( è è (33 x ( i o-iceaig i ' ' ( d - >, =,, whee d i a aiay ume uch ha ( codiio (3 ad (33 hold uifomly i x ad E ² i ( ² E, q(, mea of he eie ( ad he cojugae fucio f ( x i defied fo almo evey x y f ( x ( co( d lim h ( co( =- y = y d ç - çè (35 Noe 3 xç xç, fo ç ³ ç è è è è Noe 33 Alo fo =, Theoem 3 educe o heoem, ad hu geealize he heoem of Nigam ad Shama [8] Noe 34 The oduc afom ( E, q (, lay a imoa ole i igal heoy a a doule digial file [9] ad heoy of Machie i Mechaical Egieeig 4 Lemma Fo he oof of ou heoem, followig lemma ae equied: Lemma : ² G( =O fo < ë Poof: Fo <, i ³ ad co ² co v - çè G( q å ( q = ç çè å è v= i ë co v ç q - è å ç ç å ( q = è è v= i ë = ( åç q q ( q = ( å = åç ëè v= ( q = è h q - -
7 Aoximaio of cojugae of igal 79 ( ( = q = O, q ë å ç q = è Lemma : ² G( = O, fo ë ad ay Poof: Fo, i ³ ² co v - çè G( q å ( q = ç çè å è v= i ë - ice = ( q i v q - ìï ç ü e è ï åç ( q = ç íå ý è è ïî v= ë ïþ iv q - ì ü åç e e ( q = ç íå ý ëè è î v= þ iv q - ì ü åç e ( q = ç íå ý ëè è î v= þ - iv q - ì ü åç e ( q = ç íå ý ëè è î v= þ iv q - ì ü ç e ( q = ç í ý ëè è î v= þ Now coideig fi em of equaio (4, we have - iv q - ì ü åç e ( q = ç íå ý ëè è î v= þ i å å (4 - - iv q - ì ü e åç í ý ( q = ç å q - ëè è î v= þ ( q å ç = ëè (4 Now coideig ecod em of equaio (4 ad uig Ael lemma, we have
8 7 Vihu Naaya Miha e al iv q - ì ü åç e ( q = ç íå ý ëè è î v= þ q - åç ç max q ( è è m = v= ( q - åç ( q = ç q - = åç ëè è ( q = ëè O comiig (4, (4 ad (43, we ge ² - G( q q å q ç = å è q ç = è ² ( G =O ë - - ( ( m å e iv (43 5 Poof of Theoem Le ² ( x deoe he aial um of eie ( The followig Nigam ad Shama [8], we have ² co ( ² ç çè x - f( x = ( d y i ç çè Theefoe uig ( he (, afom of ± i give y co ± ² ç è - f( x = y( d ( å = i çè ç è Now deoig ( ² E, q (, afom of ± ² ² ( ( ² q a ( E,, we wie q - y( ì ü E, - f x = q ï co v ïd å ( q í ý ç = è çè i å ç v= è ïî ïþ ë y( G² = ( d = y( G² ( d = I I (ay (5 ë
9 Aoximaio of cojugae of igal 7 We coide, I y( G² ( d Uig Hlde Iequaliy ( ² ( x G ( y I i d d ( ç x i è çè ë ë ( ² x G ( =O d ç è i y equaio (3 çè ë x ( =O d ç ç y Lemma è ç i ë è ( ( x ( ç =Oç d è i ç ë è ç ( ç x ( =O d ç ç, a h ç i è h è ç ( ë è x( =O d ç ç è h è ë Sice x ( i a oiive iceaig fucio ad uig Secod Mea Value Theoem fo iegal, we have =O d ç ç ç ç, fo ome <Î< èè è è ë x Î I - - =O ç ç x ç d èè è Î ë
10 7 Vihu Naaya Miha e al Noe ha I xç xç, è è - - ì ü =Oç ç xç í ý èè è î- - þ Î ë - - =O ç ç xç ( =O ç xç ( èè è è è =O ( ç xç =, Q (5 è è Now, we coide, I y ( G² ( d Uig Hlde Iequaliy -d i y ( x( G ( I d d -d x( i ç è ç è ë ë ² ( G ( x d =O (( d y equaio 33 - d i ç è ë ² d x ( =O (( ç d y Lemma -d i ç ë è d (( =O d (( ë x ( ç d è - d ç i x ( =O ç - d è ë d
11 Aoximaio of cojugae of igal 73 Now uig, y = I ( ( y d-- y y ç ( x d ç dy =O ( ç ë è x ( y x i a oiive iceaig fucio o Sice ( i alo a oiive iceaig y fucio ad uig Secod Mea Value Theoem fo iegal, we have I ( ( x d dy ( =O ç, fo ome h ç ç è è ë - d y h ( ì -d - ü d y ï ï =O ç ( xç, fo ome í ý d è è ïë- - ï î þ ( d - d - ì ü =O ç ( xç í y ý ë è è î þ =O ç ç è è d -d- ( x ( d -d- =O ç xç ( =O ç xç ( =, è è è è Q (53 omiig I ad I yield ² q ( E f - =O x ç è çè (54 Now, uig he L -om of a fucio, we ge / ì ü ìï ü ï dx ï í ý dxï í x ý ç çè ïî ïþ ç èïî ïþ / ç q q ï / è E ² - f = E ² - f =O ï ( / / O ( dx / = x = O ( ç ç è çç è x çè ç çè è Thi comlee he oof of Theoem 3 6 Alicaio The heoy of aoximaio i a vey exeive field, which ha vaiou alicaio Fom he oi of view of he alicaio, Shae eimae of ifiie maice[5], ae ueful o ge oud fo he laice om (which occu i olid ae hyic of
12 74 Vihu Naaya Miha e al maix valued fucio, ad eale o iveigae euaio of maix valued fucio ad comae hem The followig coollaie may e deived fom Theoem 3 oollay 6 If x ( a, a, W L, x, ³, = < he he weighed cla ( ( educe o he cla Li(, cojugae o a - eiodic fucio f elogig o he cla Li( a,, a ad he degee of aoximaio of a fucio f( x, ² q E - f =Oç ç a- ç ( è Poof: The eul follow y eig = i (3 oollay 6 If ( x = a fo a i give y (6 < < ad i coollay (6, he f Î Lia ad ² q E - f =O a ( (6 ç è Poof: Fo i coollay (6, we ge ² ² ( q q E u - f = E x - f( x =Oç x ( a ç è Acowledgeme: The auho ae gaeful o hei eloved ae fo hei ecouageme o hei wo The auho ae highly haful o he aoymou efeee fo he caeful eadig, hei ciical ema, valuale comme ad eveal ueful uggeio which heled gealy fo he oveall imoveme ad he ee eeaio of hi ae The auho ae alo haful o all he meme of edioial oad of IJMS ad D Emil Michev, Peide of Hiai Ld, Maagig edio of Ieaioal Joual of Mahemaical Aalyi (Hiai Ld, Rue, Bulgaia, EU, fo hei id cooeaio ad mooh ehavio duig commuicaio feece [] A Zygmud, Tigoomeic eie, d ed, Vol, amidge Uiv Pe, amidge (959 [] B E Rhoade, K Ozolu ad I Alaya, O degee of aoximaio o a fucio elogig o he cla Lichiz cla y Haudoff mea of i Fouie eie, A Mah ad omuaio, 7(, [3] E Z Paai ad G V Mouaide, A L aed mehod fo he deig of - D zeo hae FIR digial file, IEEE Ta icui ad Sy I, Fudameal Theo ad Al, 44 (997, 59-6 [4] G H Hady, Divege eie, fi ed, Oxfod Uiveiy Pe, 7 (949 [5] H H Kha, O degee of aoximaio o a fucio elogig o he weighed W ( LP, x (,( ³ - cla, Aligah Bull of Mah, 3-4 ( , [6] H H Kha, O degee of aoximaio o a fucio elogig o he cla Li(α,, Idia J of Pue ad Al Mah, 5 (974 o, 3-36
13 Aoximaio of cojugae of igal 75 [7] H H Kha, A oe o a heoem of Izumi, omm Fac Mah Aaa, 3, (98, 3-7 [8] H K Nigam ad A Shama, O aoximaio of cojugae of a fucio elogig o Li( x(,, - la y oduc ummailiy mea of cojugae eie of Fouie eie, I J of oem Mah Sci, Vol 5 (, No54, [9] J G Poai, Digial ommuicaio, McGaw-Hill, New Yo, 985 [] M L Mial ad B E Rhoade, Aoximaio y maix mea of doule Fouie eie o coiuou fucio i wo vaiale fucio, Radovi Ma, 9 (999, [] M L Mial ad B E Rhoade, Degee of aoximaio of fucio i a omed ace, J of omu Aal Al ( - [] M L Mial ad B E Rhoade, Degee of aoximaio of fucio i he Hlde meic, Radovi Ma, (, 6-75 [3] M L Mial ad B E Rhoade, O he degee of aoximaio of coiuou fucio y uig liea oeao o hei Fouie eie, I J of Mah, Game Theo, ad Algea 9 (999, [4] M L Mial ad Vihu Naaya Miha, Aoximaio of igal (fucio elogig o he Weighed W ( LP, x (,( ³ - la y almo maix ummailiy mehod of i Fouie eie, I J of Mah Sci ad Egg Al, Vol (8, No IV, - 9 [5] M L Mial, B E Rhoade, S Soe, ad U Sigh, Aoximaio of igal of cla Li( a,,( ³ y liea oeao, Al Mah ad om, 7 (, [6] M L Mial, B E Rhoade ad Vihu Naaya Miha, Aoximaio of igal (fucio elogig o he weighed W ( LP, x (,( ³ - la y liea oeao, I J Mah Sci, ID (6, - [7] M L Mial, B E Rhoade, Vihu N Miha, S Pii ad S S Mial, Aoximaio of fucio elogig o Li( x (,,( ³ - la y mea of cojugae Fouie eie uig liea oeao, Id J of Mah, 47 (5, 7-9 [8] Vihu Naaya Miha, O he Degee of Aoximaio of igal (fucio elogig o Geealized Weighed W ( LP, x (,( ³ - la y almo maix ummailiy mehod of i cojugae Fouie eie, I J of Al Mah ad Mech, 5 (7(9, 6-7 [9] Vihu Naaya Miha, O he Degee of Aoximaio of igal (fucio elogig o Geealized Weighed W ( LP, x (,( ³ - la y oduc ummailiy Mehod, Joual of Ieaioal Academy of Phyical Sciece, Vol 4 (, No4, ceived: Seeme,
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