Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator

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1 Alied Mhemics doi:.4236/m Pulished Olie Decemer 2 (h:// Degree of Aroimio of Cojuge of Sigls (Fucios) y Lower Trigulr Mri Oeror Asrc Vishu Nry Mishr Huzoor H. Kh 2 Kejl Khri Derme of Mhemics S. V. Niol Isiue of Techology Sur Idi 2 Derme of Mhemics Aligrh Muslim Uiversiy Aligrh Idi E-mil: vishu_rymishr@yhoo.co.i huzoorh@yhoo.com ejl99@gmil.com Received My 4 2; revised Ocoer 25 2; cceed Novemer 5 2 I he rese er em is mde o oi he degree of roimio of cojuge of fucios (sigls) elogig o he geerlized weighed W(L P ξ()) ( )-clss y usig lower rigulr mri oeror of cojuge series of is Fourier series. Keywords: Cojuge Fourier Series Geerlized Weighed W(L P ξ())-clss Degree of Aroimio d Lower Trigulr Mri Mes. Iroducio Le f e 2 -eriodic sigl (fucio) d le f L 2 L. The he Fourier series of fucio (sigl) f y oi is give y f ( ) cos si 2 (.) u ( f; ) wih ril sums s ( f; ) rigoomeric olyomil of degree (or order) of he firs ( ) erms. The cojuge series of Fourier series (.) of f is give y ( cos si ) v( f; ) (.2) wih ril sums s ( f; ). If f is Leesgue iegrle d f Li( ( ) ) he 2 f ( ) co ( 2)d lim co ( 2)d h h eiss for ll Zygmud [. 3] f ( ) is clled he cojuge fucio of f ( ). The mri T ( ) i which is he eleme i -h row d -h colum is usully clled he mri of T. Mrices T such h for re clled lower rigulr. Le T ( ) e ifiie lower rigulr mri sisfyig Töeliz [2] codiios of regulriy i.e. of M where M is fiie cos ideede lim for ech d lim. Le u e ifiie series whose ( ) h ril sum s u. The seuece-o-seuece rsformio s s 2 defies he seuece of lower rigulr mri summiliy mes of seuece s geered y he seuece of coefficies ( ). The rsforms re clled lier mes or mri mes (deermied y he mri T) of he seuece s. A ifiie series u is sid o e summle o s y lower rigulr mri T-mehod if lim eiss d is eul o s Zygmud [. 74] d we wrie s ( T ) s The summiliy mri T or he seueceo-seuece rsformio is sid o e regulr if lim s s lim s. A fucio (sigl) f ( ) Li for if f ( ) f( ) ( ). Coyrigh 2 SciRes.

2 A fucio (sigl) f ( ) Li( ) for Fdde [3] if 2 f ( ) f( ) d ( ) Give osiive icresig fucio () d ieger f( ) Li( ( ) ) Kh [4] if 2 f ( ) f( ) d ( ( ) ). I cse () he Li ( ( ) ) coicides wih he clss Li ( ). If i Li ( ) clss he his clss reduces o Li. For give osiive icresig fucio () ieger f( ) W ( L ( )) Kh [4] if 2 f( ) f( ) si d ( ( ))( ). We oe h if he he geerlized weighed W( L ( ))( ) -clss coicide wih he clss Li( ( ) ). Also we oserve h Li Li ( ) Li ( ( ) ) W ( L ( )) for Mishr [5]. The L -orm is defied y 2 f f( ) d. V. N. MISHRA ET AL. 449 G() ()d G( ) ()d. Here G ( ) sds for lim G he eisece of which follows from he codiios. Noe h i is esseil h he iervl ( ] cois. A vri o hvig his reuireme is: If G: R is moooic (o ecessrily decresig d osiive) fucio d : R is iegrle fucio he umer such h G() ()d G( ) ()d G( ) ()d. We use he followig oios: () f( ) f( ) A A r r cos( 2) M () 2 si( 2) ieger o eceedig of. he grees Furhermore C will deoe solue osiive cos o ecessrily he sme ech occurrece. Throughou his er we e ( ) d A. 2. Mi Resul The L - orm of fucio f : R R is defied y I is well ow h he heory of roimios i.e. f su f( ) : R f which origied from well ow heorem of d he degree of roimio E ( ) Weiersrss hs ecome eciig ierdisciliry f of fucio field of sudy for he ls 3 yers. These roimf : R R is give y ios hve ssumed imor ew dimesios due o E( f) Mi f( ) ( f; ) heir wide licios i sigl lysis i geerl d i digil sigl rocessig [5] i riculr i view of i erms of where ( f; ) is rigoomeric oly- he clssicl Sho smlig heorem. omil of degree (order). This mehod of roim- This hs moived y vrious ivesigors such s io is clled rigoomeric Fourier Aroimio (f) Qureshi ([78]) Kh ([49]) Chdr [] Leidler [] Mishr [6]. Riesz-Hölder Ieuliy ses h if d Mishr [5] discussed he degree of roimio of e o-egive eeded rel umers such h sigls (fucios) elogig o. If f L d g L he Li Li( ) Li( ( ) ) d W( L ( )) -clsses y fg. L d usig Cesàro d Nörlud mes of ifiie series. Qureshi ([23]) hve deermied he degree of f ( ) fg f g. cojuge of fucio f( ) Li d Li( ) y Nörlud mes of cojuge series of Fourier series. Euliy holds if d oly if for some o-zero co- The urose of his er is o deermie he degree of ss A d B we hve A f B g.. e roimio of f ( ) cojuge of fucio Secod Me Vlue heorem for iegrio ses h f( ) W( L ( ))( ) y lower rigulr mri if G: R is osiive moooiclly decresig mes. fucio d : R is iegrle fucio We rove: he umer such h Theorem 2.. Le T ( ) e ifiie regulr Coyrigh 2 SciRes.

3 45 V. N. MISHRA ET AL. lower rigulr mri such h he elemes ( ) e o-egive o-decresig wih. If f : R R is 2 -eriodic Leesgue iegrle d el ogig o he geerlized weighed W( L ( )) -clss he he degree of roimio of f ( ) cojuge of f ( ) W( L ( )) y lower ri gulr mri mes ( f; ) is give y / f ( ) ( f; ) ( ) O (2.) rov ided () is osiive icresig fu cio of sisfyig he followig codiios ψ() si d O( ) (2.2) ξ() ψ() ) d O( ) ξ() (2.3) ξ() d is decresig i (2.4) where is rirry umer such h ( δ + ) > he cojuge ide of d co- diios (2.2) (2.3) hold uiformly i d + =. Noe. Codiio (2.4) imlies for Noe 2. Also fo r our Theorem (2.) reduces o oe of he heorem of Ll d Kushwh [4]. 3. Lemms I order o rove our Theorem 2. we reuire he followig lemm. Lemm 3.. Uder he codiios of our Theorem 2. o ( ) we hve A M () O for. Proof. For (si ) 2for 2 we hve M () 2 si( 2) 2 si( 2) cos( 2) cos( 2) 2 2 cos( 2) cos( 2) r 2 m cos( 2) 2 r O 2 cos( 2) A d A ( ) A Therefore M () O This comlees he roof of Lemm Proof of Theorem 2. h The ril sum of he cojuge series of he Fourier series (.2) is give y The or s ( f; ) co( 2) ( )d 2 cos( 2) ()d 2 si( 2) s ( f; ) co( 2) ( )d 2 cos( 2) 2 ()d si( 2) 2 s ( f; ) co( 2) ( )d cos( 2) ()d 2 si( 2) ( f ; ) f ( ) II2 (4.) Usig Riesz-Hölder s ieuliy codiio (2.2) (2.4) oe he fc h (si ) for 2 2 d he secod me vlue heorem for iegrls we fid Coyrigh 2 SciRes.

4 V. N. MISHRA ET AL. 45 () I si d () () M () d si () si d () () cos( 2) si si ( 2) ( ) O O 2 si d d () O O d ; h 2 si( ) h ( ) O d ; h 2 h (2 ) O O d h 2 O O / O (4.2) Now y Riesz-Hölder s ieuliy codiios (2.3) (2.4) oe Lemm 3. he fc h (si ) for I () M ()d we oi ( )si M() d d () si ( ) si () d () A O d si O () A d / Sice A hs o-egive eries d row sums oe ( y) dy O 2 y y ( ) dy O 2 y ( ) ( ) O ( ) O O O ( ) O ( ). (4.3) Comiig I d I 2 yields ( f; ) f ( ) O ( ). Now usig he L -orm we ge 2 ( f; ) f( ) ( ; ) ( ) d f f 2 / O ( d 2 O ( ) d O ( ). This comlees he roof of our Theorem Alicios The followig corollries c e derived from our Theo- rem 2.. Corollry 5.. If d () he he geerlized weighed clss W L ( ) reduces o clss Li ( ) d he degree of roimio of fucio f ( ) Li( ) is give y ( ; ) ( ) f f O. Proof of corollry 5.. From our Theorem 2. for we hve 2 ( f ; ) f( ) ( ; ) ( ) d f f O ( ) O. This comlees he roof of corollry 5.. Corollry 5.2. If i corollry 5. he for Coyrigh 2 SciRes.

5 452 V. N. MISHRA ET AL. f f O ( ; ) ( ). Corollry 5.3. If P P ( ) he he degree of roimio of f ( ) cojuge of f Li( ) y Nörlud mes ( f ; ) P ( ; ) s f of he cojuge series of Fourier series is give y ( f; ) f ( ) O. Co rollry 5.4. If P P ( ) he he degree of roimio of f ( ) cojuge of f Li y Nörlud mes P s of he cojuge series of Fourier series is give y f O ( ) (5.) Olog( ) e ( ). Corollry 5.5. If R such h Ryy R is moooic o-decresig he he degree of roimio of f ( ) cojuge of fucio f Li y geerlized Nörlud ( f ; ) R ( ; ) s f mes of he cojuge series (.2) sisfies euio (5.). 6. Remrs R emr 6.. Ll d Kushwh [4]. The degree of - roimio ( f; ) f ( ) O. deermied y Qureshi [3. 56 L. 2] eds o if 3 d 2 d lso for oher vlues. Therefore his deficiecy hs ecourged o ivesige degree of roim io of cojuge of fucios e- logig o Li ( ) cosiderig. 7. Acowledgemes The uhors re greful o his eloved res for heir ecourgeme o his wor. The uhors re greful o he referee for his vlule suggesios d useful commes for he imroveme of his er. The uhors re lso hful o he Edior i chief Prof. Chris C- igs Uiversiy of Sheffield UK d Edioril Assis Ms. Ti Hug Scieific Reserch Pulishig USA for heir id cooerio durig commuicio. 8. Refereces [2] O. Töeliz Üer Allgemeie Liere Mielilduge Prce Memyczo Fizycze Jourl Vol h:// h/?=359 [3] L. McFdde Asolue Nörlud Summiliy Due Mhemicl Jourl Vol doi:.25/s x [4] H. H. Kh O he Degree of Aroimio o Fucio Belogig o Weighed (L P ξ()) Clss Aligrh Bullei of Mhemics Vol [5] V. N. Mishr Some Prolems o Aroimios of Fucios i Bch Sces Ph.D. Thesis Idi Isiue of Techology Rooree 27. [6] V. N. Mishr O he Degree of Aroimio of Sigls (Fucios) Belogig o he Weighed W(L P ξ()) ( )-Clss y Almos Mri Summiliy Mehod of Is Cojuge Fourier Series Ieriol Jourl of Alied Mhemics d Mechics Vol. 5 No [7] K. Qureshi O he Degree of Aroimio of Fucio Belogig o Liα Idi Jourl of Pure d A- Degree of Aroimio of Fuc- lied Mhemics Vol. 3 No [8] K. Qureshi O he io Belogig o he Clss Li(α) Idi Jourl of Pure d Alied Mhemics Vol. 3 No [9] H. H. Kh O he Degree of Aroimio o Fucio y Trigulr Mri of Is Cojuge Fourier Series II Idi Jourl of Pure d Alied Mhemics Vol [] P. Chdr Trigoomeric Aroimio of Fucios i L P -Norm Jourl of Mhemicl Alysis d Alicios Vol doi:.6/s22-247x(2)2- [] L. Leidler Trigoomeric Aroimio i L P -Norm Jourl of Mhemicl Alysis d Alicios Vol doi:.6/j.jm [2] K. Qureshi O he Degree of Aroimio of Cojuge of Fucio Belogig o he Lischiz Clss y Mes of Cojuge Series Idi Jourl of Pure d Alied Mhemics Vol. 2 No [3] K. Qureshi O he Degree of Aroimio of Cojuge of Fucio Belogig o he Clss Li(α) y Mes of Cojuge Series Idi Jourl of Pure d Alied Mhemics Vol. 3 No [4] S. Ll d J. K. Kushwh Aroimio of Cojuge of Fucios Belogig o he Geerlized Lischiz clss y Lower Trigulr Mri Mes Ieriol Jourl of Mhemicl Alysis Vol. 3 No [] A. Zygmud Trigoomeric Series Vol. I Cmridge Uiversiy Press Cmridge 959. Coyrigh 2 SciRes.