Trigonometric Approximation of Signals. (Functions) in L p -Norm

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1 It. J. Cote. Mth. Scieces, Vol. 7, 202, o. 9, Trigooetric Aroxitio of Sigls (Fuctios) i L -Nor Vishu Nry Mishr d Lshi Nry Mishr 2 Dertet of Mthetics, S.V. Ntiol Istitute of Techology, Ichchhth Mhdev Rod, Surt, Surt (Gujrt) Idi vishu_ryishr@yhoo.co.i, vishuryishr@gil.co 2 Dr. R Mohr Lohi Avdh Uiversity, Fizbd (U.P.), Idi l ishr@yhoo.co.i, lshiryishr04@gil.co Abstrct Brodly seig, Sigls re treted s fuctios of oe vrible d iges re rereseted by fuctios of two vribles. The study of these cocets is directly relted to the eergig re of ifortio techology. I this er, we exted two theores of Leidler [J. Mth. Al. Al. 302 (2005) 29-36], where he hs te less striget coditios o the geertig sequece give by Chdr [J. Mth. Al. Al. 275, 2002, 3-26], to ore geerl clsses of trigulr trix ethods. Our Theores lso geerlize theore 4 rtilly d (ii) rt of theore 5 of M.L. Mittl, B.E. Rhodes, V.N. Mishr, U. Sigh [J. Mth. Al. Al. 326(2007) ] by droig the ootoicity o the eleets of trix rows.. Itroductio Let f be 2π -eriodic sigl (fuctio) d let f L 0, 2π L for. The the Fourier series of fuctio (sigl) f t y oit x is give by 0 f( x) cos x b si x u(f ; x), 2 (.) 0 with rtil sus s(f;x) - trigooetric olyoil of degree(or order), of the first ( ) ters. We defie where T(),, 0 (f;x) s(f;x), 0, (.2) is lier oertor rereseted by the ifiite lower trigulr trix. The series (.2) is sid to be T suble to s, if (f ; x) s s.

2 90 Vishu Nry Mishr d Lshi Nry Mishr The T oertor reduces to the Nörlud (N) -oertor, if P, 0,,, where P 0 d 0,, 0 P. I this 0 cse, the trsfor (f ; x) reduces to the Nörlud trsfor N(f ; x). The T oertor reduces to the weighted (Riesz) () - oertor, if P, 0,,, where P 0 d 0,, 0 P. I this 0 cse, the trsfor (f ; x) reduces to the weighted (Riesz) trsfor (f ; x) or R(f ; x). A sigl (fuctio) f Li, for 0, if f(x t) f(x)(t). A sigl (fuctio) f Li (,) for, 0, if 2 0 / f(x t) f(x) dx(t). The itegrl odulus of cotiuity of fuctio If, for α 0, Throughout Let T, f L [0, 2 ] is defied by 2 π (δ; f) su f(x h) f(x) dx. 0 h δ 2π 0 α (δ; f) O(δ), the f Li(, ) ( ).. will deote L -or with resect to x d will be defied by 2 π f f(x) dx f L( ). 2π 0 be lower trigulr regulr trix with o-egtive etries. A trix T is sid to hve ootoe rows if, for ech,, is either oicresig or o-decresig i, 0. A lower trigulr trix T is clled hu trix if, for ech, there exists iteger 0 0 (), such tht,, for 0 0, d,, for. 0 A ositive sequece c: c / is clled lost ootoe decresig ( or icresig) if there exists costt K: K(c), deedig o the sequece c oly, such tht for ll, c K c(k c c). Such sequeces will be deoted by ca M DS d ca M IS, resectively. D(t) si / 2 t / 2si t / 2, is ow s Dirichlet Kerel of We write degree (or order). /

3 Trigooetric roxitio of sigls 9 2 π s(f) f(x t) D(t) dt,(f ; x) s(f ; x), π 0 0 N(f;x) s(f;x), P 0, 0 P. r P 0 r 0 R(f;x) s(f;x), P 0, 0 P. r P 0 r 0 A, A, 0 A,, r, t, A,0, b,, r 0 Δ, [x]- deotes the gretest iteger cotied i x.,,, I this er, the sigls (fuctios) f re roxited by trigooetric olyoils of order (or degree) d the degree of roxitio E( f) is give by E(f) Mi f(x)(f;x), i ters of. This ethod of roxitio is clled trigooetric Fourier Aroxitio (tf) d used i the theory of Mchies i Mechicl Egieerig. Let (f) deote the th ter of the (C, ) trsfor of the rtil sus of the Fourier series of 2-eriodic fuctio f. I 937, Qude [] hs exteded the results for the fuctios f Liα, for 0 α, o Cesàro trix: f Li α, for 0 α, the Theore []. If f(x)(f ; x) O() (.4) for either (i) d 0 α or (ii) d 0 α. d if α, the f(x)(f ; x) O log( ). (.5) I recet er Chdr [4] exteded the wor of Qude [] d roved the followig theores, where N(f) d R(f) deote the th ters of the Nörlud d weighted e trsfors of the sequeces of rtil sus, resectively. Theore 2 [4]. Let f Li(,) be ositive sequece such tht d let O P. (.6) If either (i), 0 d (ii) is ootoic, or (i), 0 d (ii) is o-decresig sequece, the α f(x) N(f ; x) O(). (.7) Theore 3 [4]. Let f Li(,) d let be ositive. Suose tht either (i), 0 α, d (ii) (i), 0 α 0 d (ii) P P O, or with (.6) is ositive d o-decresig. The f(x) R(f ; x) O( ). (.8)

4 92 Vishu Nry Mishr d Lshi Nry Mishr Theore 4 [4]. Let f Li(,) d let with (.6) be ositive d tht η ( ) be o-decresig for soe η 0. (.9) The f(x) R(f ; x) O(). (.0) These results of Chdr [4], s etioed by hi, re shrer d ew d hece re iterestig (i view of Leidler [2,.30] lso). Recetly, Leidler [2] hs exteded theore 2 d theore 3 by tig less striget coditios o the geertig sequece. He roved: Theore 5 [2]. Let f Li(, ) be ositive. If oe of the coditios d let (i), 0 d A M DS, (ii), 0, A M IS d (.6) holds, (iii), d (iv) (v) 0 O(P),,, O( P / ), 0, O( P / ) f(x)(f ; x) Theore 6 [2]. Let f Li(,),0 < <. (.6) d the coditio d (.6) holds, O(). itis, the If the ositive sequece 0 O( P / ) holds, the stisfies f(x) R(f ; x) O(). (.) Very recetly M.L. Mittl, B.E. Rhodes, V.N. Mishr, U. Sigh [3] hve geerlized two theores 2 d 3 (Chdr [4, Theores d 2]) to ore geerl clsses of trigulr trix ethods. They rove: Theore 7 [3]. Let f Li(,) d let T hve ootoe rows d stisfy (i) If, 0, d T lso stisfies t O(). (.2),0, r ( ) x, O(), (.3) where r: [ / 2], the (f ; x) f(x) = O(). (.4) (ii) If >, =, the (.4) is stisfied. (iii) If, 0, d T lso stisfies the (.4) is stisfied., 0, ( ) x, O(), (.5) Theore 8 [3]. Let T be hu trix stisfyig coditio (.2) d

5 Trigooetric roxitio of sigls 93, ( ) x O(). (.6) The, if either (i), 0, or (ii), 0, coditio (.4) is stisfied. I this er, we geerlize two theores 5 d 6 of Leidler [2, Theores d 2] to ore geerl clsses of trigulr trix ethods. Our Theores lso geerlize rtilly theore 7 d (ii) rt of theore 8 of M.L. Mittl, B.E. Rhodes, V.N. Mishr, U. Sigh [3] resectively by droig the ootoicity o the eleets of trix rows. We rove: Theore 9. Let f Li (,), d let T (), be ifiite trigulr trix with ositive etries (), with row sus such tht ( )(). O (.7) If oe of the coditios (i), 0,, (ii), 0,,, 0 AMDS d (.7) holds, AMIS d (.7) holds, (iii), d (iv),, ()(),, 0 O (.8), (),0 0 O d (.7) holds, (.9) 0 (v), 0,, O (), 0 d (.7) holds, (.20) itis, the (.4) is stisfied. Theore 0. Let (, ), 0 < <. f Li If the ositive sequece, stisfies coditio (.20), the ( f ;)() x f=(). x O (.2) We ote tht: () I cse of Nörlud ) - trsfor, coditio (.7) reduces to (.6), while coditios (.8) d (.9) reduce to coditios (iii) d (iv) of theore 5 resectively. Thus our Theore 9 geerlizes theore 5. Siilrly, i cse of Riesz (R) - trsfor, Theore 0 exteds theore 6. (2) Further, it is esy to exie tht the coditios of Theores 9 d 0 cli less th the requireets of our Theores 7 d 8 resectively for A, 0 t. For exle, the coditio o the su i (.8) is lwys stisfied if the sequece is o-decresig i, the usig (.7), we get, ( ) ( ) ( )( ),,,,, A( ) O() O().,0,0

6 94 Vishu Nry Mishr d Lshi Nry Mishr If, is o-icresig i d (.7) holds the ( ) O(),,,,,,0,, is lso true. 2. Les I order to rove our Theores 9 d 0, we require the followig les. Le []. If f Li(, ), 0, the f(x)(f ; x) O(). (2.) Le 2 []. If f Li(, ),, the σ(f ; x) s(f ; x) O(). (2.2) Le 3 []. Let, for 0 d, f Li(, ). The f(x) s(f ; x) O(). (2.3) Le 4 [3]. Let T hve ootoe rows d stisfy (.3). The, for 0, 3. Proof of the Theores (, ) O( ). α (2.4) 0 3. Proof of Theore 9. Cses I d II. If, 0, we rove the cses (i) d (ii) s iulteously. The roof rus siilr to the cse (i) of Theore 7, droig the secod ter s t A,0,. Let, be either AMDS or AMIS. Thus for the se of coleteess, we hve thus (3.) (f; x) f(x) s(f; x) f(x) s(f; x) f(x),,, 0 0 (f; x) f(x) s(f; x) f(x) O() O(),,, 0 0 i view of Les (3) d (4). Next, we cosider the cse (iv). Cse IV. If, α, we hve (f; x) f(x)(f; x) s(f; x) s(f; x) f(x). Now, usig Le 3, we get (f ; x) f(x)(f ; x) s(f; x) s(f ; x) f(x) (f ; x) s(f ; x) O().(3.2) Now to rove our theore, it reis to show tht (f ; x) s(f ; x) O(). (3.3)

7 Trigooetric roxitio of sigls 95 Now, we write (f ; x),s(f ; x), u(f i ; x) A, u(f ; x), 0 0 i0 0 d thus, s A,0, we hve (f ; x) s(f ; x) A u(f ; x) u(f ; x)., 0 0 A, A, 0 u(f ; x) b, u(f ; x). Hece by Abel s trsfortio, we obti (f ; x) s(f ; x) b, ju(f j; x) b, ju(f ; jx). j j Thus by trigle iequlity, we fid (3.4) (f ; x) s(f ; x) b ju(f ; x) b ju(f ; x)., j, j j j Now (f;x) s(f ; x) s(f ; x) s(f ; x) 0 s(f ; x) u(f ; x) ju(f;x). j 0 0 j Therefore by Le 2, we hve j ju(f ; x)( )(f;x) s(f ; x)( )O() O(). j We ote tht A, A A, 0 A, 0, A, 0 A, A, 0 b, O(/ ). Thus (3.5), j, j (3.6) j j b ju(f;x) b ju(f;x) O(). As i cse (ii) of roof of Theore 7, we write b(, ).,, r ( ) (3.7) r0 Next we shll verify by theticl iductio tht If, the ( ) (r ). (3.8),r,,r,r r 0 r 0 r 0 2., r,,0, Thus (3.8) holds. Now let us suose tht (3.8) holds for i.e.

8 96 Vishu Nry Mishr d Lshi Nry Mishr ( ) (r ), (3.9),r,,r,r r 0 r 0 d we hve to show tht (3.8) is true for. For ( ) d usig (3.9), we get ( 2) ( ),r,,r, r 0 r 0 ( )( )( ) r 0, r,,, ( ) (r ) ( ) (r ),, r, r,,, r, r r 0 r 0 which shows tht (3.8) is true for. Thus (3.8) holds N. Usig (.7), (3.7) d (3.8), we fid b (A A ( )( ),,,0,,r r 0 ( ) (,r ) r 0, ( )(r ),r,r (),, r 0,, O(),0 O().(3.0) ( ) 0 Cobiig ( 3.4), ( 3.5), ( 3.6) d ( 3.0) yields ( 3.3). Cosequetly, usig ( 3.3) fro (3.2), we obti (f; x) f(x) O(). This coletes the roof of cse (iv). Cse III. If,. Now to rove this cse first of ll we rove tht the coditio 0 ( ) O() ilies tht, (3.),,,0 B b (A A O(). For this, usig (3.8) s i cse (iv), we hve,,r,r,r r 0 r 0 B ( )( ) ( )(r ) Now, usig (.8) d iterchgig the order of sutio, we get r ( ), B B, 2sy. (3.2) r

9 Trigooetric roxitio of sigls 97 r r,, B ( ) ( ) r,,, r r r (, ) O(), O().(3.3) r r r 0 O the other hd B ( ) ( ), sy.(3.4) r 2,, r r r,,2 Now, usig (.8), we obti r,, r B ( ) r ( ) (, ), r r r ( ) (, ), r r 2 r 2 r 2 r r 2 r 2 ( )( ) ( ), O() O(/ ), r 0 r (3.5) gi usig (.8) d iterchgig the order of sutio, we hve,2,, r r r r B ( ) ( ) ( ) r r,, r r ( ), O() O().(3.6) r r r Fro (3.2), (3.3), (3.4), (3.5) d (3.6), we get (3.). Thus (3.2), (3.4) d Le 3 gi yield (.4). Cse V. If, 0, usig (3.),, 0 d the Abel s trsfortio, we obti (f ; x) f(x)( ), s(f ; x) f(x)( r ) s(f ; x), f(x), r 0 r 0 r 0 ( ), s(f ; rx) f(x)( )( )(f ; x),f(x). 0 r 0 0 Hece, by coditio (.7), (.20) d Le, we fid

10 98 Vishu Nry Mishr d Lshi Nry Mishr (f ; x) f(x)( ) (f ; x) f(x), O( ), 0 0,,0 0 O() O() O() O() O() O(). This coletes the roof of cse (v) d cosequetly the roof of Theore 9 is colete. 3.2 Proof of Theore 0 The roof is exctly the se s the roof of cse (v) of Theore 9. If we write, / P i the roof of cse (v) of Theore 9, we get the roof of theore 6 d hece the roof of cse (v) of Theore 9 will be se s the roof of Theore 0. Acowledgeets The uthors re grteful to their beloved rets for their ecourgeet to their wor. The uthor is thful to his Ph.D. suervisor for fruitful discussio of the er durig the eriod of reserch t IIT, Rooree d this er is rt of Vishu Nry Mishr s Ph.D. Thesis [5]. The uthors wish to exress their grtitude to the referee for his detiled criticis d elborte suggestios which hve heled the to irove the er substtilly. They hve thus bee ble to eliite soe istes d to reset the er i ore coct er. The uthor is lso thful to ll the ebers of editoril bord of IJCMS d Dr. Eil Michev, Presidet of Hiri Ltd., Mgig editor of IJCMS, for their id cooertio durig couictio d lso uch thful to Reserch Scholr Kejl Khtri for tye settig, rerrgeet of ters, Clericl wor d rerig the.df file etc of the er. Refereces []. E.S. Qude, Trigooetric roxitio i the e, Due Mth. J. 3 (937) [2]. L. Leidler, Trigooetric roxitio i L -or, J. Mth. Al. Al. 302 (2005) [3]. M. L. Mittl, B. E. Rhodes, V. N. Mishr d Udy Sigh, Usig ifiite trices to roxite fuctios of clss Li(, ) usig trigooetric olyoils, J. Mth. Al. Al. 326 (2007) [4]. P. Chdr, Trigooetric roxitio of fuctios i L -or, J. Mth. Al. Al. 275 (2002) [5]. Vishu Nry Mishr, Soe Probles o Aroxitios of Fuctios i Bch Sces, Ph.D. Thesis (2007), Idi Istitute of Techology, Rooree, Rooree , Uttrhd, Idi. Received: October, 20