Research Article Approximation of Signals (Functions) by Trigonometric Polynomials in L p -Norm

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1 Intenational Matheatics and Matheatical Sciences, Aticle ID , 6 pages Reseach Aticle Appoxiation of Signals (Functions) by Tigonoetic Polynoials in L p -No M. L. Mittal and Madul Vee Singh Depatent of Matheatics, Indian Institute of Technology Rookee, Rookee, Uttaakhand , India Coespondence should be addessed to Madul Vee Singh; adul.singh@gail.co Received 3 Januay 24; Revised 3 Mach 24; Accepted 3 Mach 24; Published 9 Apil 24 AcadeicEdito:A.Zayed Copyight 24 M. L. Mittal and M. V. Singh. This is an open access aticle distibuted unde the Ceative Coons Attibution License, which peits unesticted use, distibution, and epoduction in any ediu, povided the oiginal wok is popely cited. Mittal and Rhoades (999, 2) and Mittal et al. (2) have initiated a study of eo estiates E n (f) though tigonoetic- Fouie appoxiation (tfa) fo the situations in which the suability atix T does not have onotone ows. In this pape, the fist autho continues the wok in the diection fo T to be a N p -atix. We extend two theoes on suability atix N p of Degeetal.(22)wheetheyhaveextendedtwotheoesofChanda(22)usingC λ -ethod obtained by deleting a set of ows fo Cesào atix C. Ou theoes also genealize two theoes of Leindle (25) to N p -atix which in tun genealize the esult of Chanda (22) and Quade (937). IneoyofPofessoK.V.Mital, Intoduction Let f be a 2π peiodic signal (function) and let f L p : L p [, 2π], p.let s n (f) : s n (f; x) a n 2 + n k u k (f; x) (a k cos kx+b k sin kx) denote the patial sus, called tigonoetic polynoials of degee (o ode) n, ofthefist(n + ) tes of the Fouie seies of f at a point x. The integal odulus of continuity of f is defined by ω p (δ; f) : sup { < h δ If, fo α>, 2π 2π () f (x+h) f(x) p /p dx}. (2) ω p (δ; f) O (δ α ), (3) then f Lip(α, p) (p ).Thoughout p will denote the L p -no, defined by f p : { 2π 2π f (x) p /p dx}, f L p (p ). (4) Apositivesequencec : {c n } is called alost onotone deceasing (inceasing) if thee exists a constant K:K(c), depending on the sequence c only, such that, fo all n, c n Kc (Kc n c ). (5) Such sequences will be denoted by c AMDS and c AMIS, espectively. A sequence which is eithe AMDS o AMIS is called alost onotone sequence and will be denoted by c AMS. Let F be an infinite subset of N and F as the ange of stictly inceasing sequence of positive integes; say F {} n.thecesào subethod C λ is defined as (C λ x) n x λ (n) k, (n, 2, 3,...), (6)

2 2 Intenational Matheatics and Matheatical Sciences whee {x k } is a sequence of eal o coplex nubes. Theefoe, the C λ -ethod yields a subsequence of the Cesào ethod C, and hence it is egula fo any λ. C λ is obtained by deleting a set of ows fo Cesào atix. The basic popeties of C λ -ethodcanbefoundin[, 2]. In the pesent pape, we will conside appoxiation of f L p by tigonoetic polynoials (f; x) and Rλ n (f; x) of degee (o ode) n,whee R λ n (f; x) (f; x) P P p s (f; x), p s (f; x), s n (f; x) 2π π f (x+t) D n (t) dt, (sin (n+/2) t) D n (t), 2 sin (t/2) P p +p + +p (n ), and by convention p P. The case p n fo all n( )of eithe (f; x) o (f; x) yields R λ n (7) σ λ n (f; x) s λ (n) + (f; x). (8) We also use Δa n a n a n+, Δ g (n, ) g(n, ) g(n, + ). (9) Mittal and Rhoades [3, 4]haveinitiatedthestudyofeo estiates E n (f) though tigonoetic-fouie appoxiation (tfa) fo the situations in which the suability atix T does not have onotone ows. In this pape, the fist autho continues the wok in the diection fo T to be a N p - atix. Recently, Chanda [5]haspovedtheetheoeson the tigonoetic appoxiation using N p -atix. Soe of the give shape estiates than the esults poved by Quade [6], Mohapata and Russell [7], and hiself ealie [8]. These esults of Chanda [5] ae ipoved in diffeent diections by diffeent investigatos such as Leindle [9] whodopped the onotonicity on geneating sequence {p n } and Mittal et al. [, ] who used oe geneal atix while vey ecently Dege et al. [2] used oe geneal C λ -ethod in view of Aitage and Maddox []. 2. Known Results Leindle [9]povedthefollowing. Theoe (see [9]). If f Lip(α, p) and {p n } be positive. If one of the conditions (i) p>, <α<,and{p n } AMDS, (ii) p>, <α<,and{p n } AMIS, (n+) p n O(P n ) holds, () (iii) p>, α,and n k Δp k O(P n ), (iv) p >, α, n k Δp k O(P n /n),and() holds, (v) p, <α<,and n k Δp k O(P n /n) aintains, then f N n(f) p O(n α ). () Theoe 2 (see [9]). Let f Lip(α, ), < α <. If the positive {p n } satisfies conditions () and n k Δp k O(P n /n) hold, then Dege et al. [2]poved. f R n(f) O(n α ). (2) Theoe 3 (see [2]). Let f Lip(α, p) and let {p n } be positive such that (λ (n) +) p O(P ). (3) If eithe (i) p>,<α,and{p n } is onotonic o (ii) p, <α<,and{p n } is nondeceasing, then f Nλ n (f) p O(n α ). (4) Theoe 4 (see [2]). Let f Lip(α,),<α<.Ifthe positive {p n } satisfies condition (3) and is nondeceasing, then 3. Main Results f Rλ n (f) O(n α ). (5) In this pape we genealize Theoes 3 and 4 of Dege et al. [2], by dopping onotonicity on the eleents of the atix ows which in tun genealize Theoes and 2, espectively, of Leindle [9] to a oe geneal C λ -ethod. We pove the following. Theoe 5. If f Lip(α, p) and {p n }is positive and if one of the following conditions (i) p>, <α<,and{p n } AMDS, (ii) p>, <α<, {p n } AMIS, and (3) holds, (iii) p>, α,and k Δp k O(P ), (iv) p >, α, k Δp k O(P /),and(3) holds, (v) p, <α<,and k Δp k O(P /) aintains, then f Nλ n (f) p O((λ (n)) α ). (6)

3 Intenational Matheatics and Matheatical Sciences 3 Theoe 6. Let f Lip(α,), <α<.ifthepositive{p n } satisfies (3) and the condition k Δp k O(P /) holds, then f Rλ n (f) O((λ (n)) α ). (7) Reaks. () If n,thenoutheoes5 and 6 educe to Theoes and 2,espectively. (2) Degeetal.[2] haveusedonotonesequences{p n } in Theoes 3 and 4, while ou Theoes 5 and 6 clai less than the equieents of thei theoes. Fo exaple, the condition of the su in (iii) of Theoe 5 is always satisfied if the sequence {p n } is noninceasing; that is, k Δp k k(p k p k+ ) P (λ (n) ) p O(P ), (8) while if sequence {p n } is nondeceasing and condition (3) holds, then the condition in (iv) of Theoe 5 is also satisfied; that is, k Δp k (p k+ p k )p p p k O( P λ (n) ). (9) Thus ou theoes genealize the two theoes of Dege et al. [2] unde weake assuptions and give shape estiates because all the estiates of Dege et al. [2] ae in tes of n, while ou estiates ae in tes of and () α n α fo <α. 4. Leas We will use the following leas in the poof of ou theoes. Lea (see [6]). If f Lip(α, p),fo<αand p>, then f s n(f) p O(n α ). (2) Lea 2 (see [6]). If f Lip(, p),fop>,then σ n(f) s n (f) p O(n ). (2) Lea 3 (see [6]). If f Lip(α, ), < α <,then f σ n(f) O(n α ). (22) Lea 4. Let {p n } AMDS o let {p n } AMIS and satisfy (3).Then,fo<α<, holds. α p O((λ (n)) α P ) (23) Poof. Let denote the integal pat of (/2). Then,if {p n } AMDS, α p α p + Kp Kp + α p α α + (+) + α + (+) α p p Kp (λ (n)) α +O((λ (n)) α P ) O((λ (n)) α P ). If {p n } AMIS and (3)holds,then α p α p + Kp + α p α α + (+) + p (24) O( P λ (n) + ) α +O((λ (n)) α P ) O((λ (n)) α P ). This copletes the poof of Lea Poof of the Main Results (25) Poof of Theoe 5. We pove cases (i) and (ii) togethe. Since (f; x) f (x) P p {s (f; x) f (x)}, (26) thus in view of Leas and 4 and condition (3), we have f Nλ n (f) p P P p f s (f) p p f s (f) p + p P s f p P p O( α )+O( p ) P O((λ (n)) α ). (27)

4 4 Intenational Matheatics and Matheatical Sciences Next we conside case (iv). Let p>and α. By Abel s tansfoation, we get and thus (f; x) s λ n (f; x) Nλ n (f; x) P P P u (f; x), (28) (P P ) u (f; x). Hence again by Abel s tansfoation, we get s λ n (f; x) Nλ n (f; x) Thus sλ n (f) Nλ n (f) p Since thus by Lea 2 n P P (29) Δ ( (P P )) ku k (f; x) + ku k (f; x). ku k (f) (λ (n) +) (3) Δ ( (P P )) p + ku (λ (n) +) k. (f) p (3) s n (f; x) σ n (f; x) ku n+ k (f; x), (32) n ku kp (n+) σ n(f) s n (f) O(). p (33) In view of (3)and(33), we obtain sλ n (f) Nλ n (f) p O( ) P Δ ( (P P )) +O((λ (n)) ). (34) Now Δ ( P P ) Δ (P P ) + P P (+) P P + P P (+) P P (+) Δ ( P P ) (+) p (+) [P P [ k Next we will veify by the induction that If, (+) p ], (35) p k (+) p ]. (36) k (+) p k p (37) k p k+ p k. k 2p k p p p. (38) Thus (37)holdsfo. Now let us assue that (37)is tue fo jand we veify j + ( ).Since k (j+2)p (j+) k (j+)p k (j+)p (j+) k jp k j p k (j+)p j +(j+)p j (j+)p (j+)

5 Intenational Matheatics and Matheatical Sciences 5 k (j+)p j k jp + (j + ) p j (j+)p (j+) j k p k+ p k +(j+) p j p (j+) j+ k p k+ p k, (39) thus (37) ispovedfoj+;thatis,(37) istuefoany. Using(36) and(37) and intechanging the ode of suation, we get Δ ( P P ) (+) k p k+ p k k p k+ p k k (+) Δp k. k (4) Now cobining this with the assuption k Δp k O(P /),wegetfo(34) sλ n (f) Nλ n (f) p O((λ (n)) ). (4) This and Lea with αyield f Nλ n (f) p O((λ (n)) ). (42) In the poof of case (iii), we fist veify that the condition k Δp k O(P ) iplies that Δ ( P P )O( P ). (43) λ (n) In view of (36)and(37) Δ ( P P ) k (+) Δ kp k + : B +B 2, say. + (44) Denoting again by the integal pat of (/2),then,by Abel s tansfoation, we have B k (+) Δ kp k Δ kp k Δp i i 2 O( P λ (n) ), (45) at the last step; we have used the condition k Δp k O(P ). Conside the following: B [ k (+) Δp k (+) : B 2 +B 22. k Δp k + k Δp k ] k Futheoe, using again ou assuption, we get B 2 B 22 + Δp i O(P i 2 λ (n) ), + Δp k k O( λ (n) )[ Δp +2 Δp + +(+) Δp + ] (46) O( P λ (n) ). (47) Suing up ou patial esults, we veified (43). Thus (34)andLea again yield f Nλ n (f) p O((λ (n)) ). (48)

6 6 Intenational Matheatics and Matheatical Sciences Now,wepovecase(v),byusing(26), p,andabel s tansfoation (f; x) f (x) P P P p {s (f; x) f (x)} (Δ p ) k (+) (Δ p ) {σ (f; x) f (x)}. HenceinviewofLea 3 Nλ n (f) f P {s k (f; x) f (x)} (+) Δ p f σ (f) O( ) (+) α P Δ p O( α ) P Δp O( α ) O( P P λ (n) ) O((λ (n)) α ). (49) (5) Heewith case (v) is also veified and thus the poof of Theoe 5 is coplete. Poof of Theoe 6. Since R λ n (f; x) (/P ) p s (f ;x), so, in view of the assuptions of Theoe 6,weget f Rλ n (f) p P {f s (f)} P (+) Δp f σ (f) + (λ (n) +) p P f σλ n (f) Conflict of Inteests The authos declae that thee is no conflict of inteests egading the publication of this pape. Acknowledgent The authos ae thankful to the leaned efeee D. A. I. Zayed fo his valuable coents and suggestions to ipove the pesentation of the pape. Refeences [] D. H. Aitage and I. J. Maddox, A new type of Cesào ean, Analysis,vol.9,no.-2,pp.95 24,989. [2] J. A. Osikiewicz, Equivalence esults fo Cesào subethods, Analysis,vol.2,no.,pp.35 43,2. [3] M. L. Mittal and B. E. Rhoades, On the degee of appoxiation of continuous functions by using linea opeatos on thei Fouie seies, Intenational Matheatics, Gae Theoy, and Algeba,vol.9,no.4,pp ,999. [4] M. L. Mittal and B. E. Rhoades, Degee of appoxiation to functions in a noed space, Coputational Analysis and Applications,vol.2,no.,pp.,2. [5] P. Chanda, Tigonoetic appoxiation of functions in L p - no, Matheatical Analysis and Applications, vol. 275,no.,pp.3 26,22. [6] E. S. Quade, Tigonoetic appoxiation in the ean, Duke Matheatical Jounal,vol.3,no.3,pp ,937. [7]R.N.MohapataandD.C.Russell, Soediectandinvese theoes in appoxiation of functions, the Austalian Matheatical Society A,vol.34,no.2,pp.43 54,983. [8]P.Chanda, AnoteondegeeofappoxiationbyNölund and Riesz opeatos, Mateatički Vesnik,vol.42,no.,pp.9, 99. [9] L. Leindle, Tigonoetic appoxiation in L p -no, Jounal of Matheatical Analysis and Applications, vol.32,no., pp.29 36,25. [] M.L.Mittal,B.E.Rhoades,V.N.Misha,andU.Singh, Using infinite atices to appoxiate functions of class Lip(α, p) using tigonoetic polynoials, Matheatical Analysis and Applications,vol.326,no.,pp ,27. [] M. L. Mittal, B. E. Rhoades, S. Sonke, and U. Singh, Appoxiation of signals of class Lip(α, p) by linea opeatos, Applied Matheatics and Coputation, vol.27,no.9,pp , 2. [2] U. Dege, İ. Dagadu, and M. Küçükaslan, Appoxiation by tigonoetic polynoials to functions in L p -no, Poceedings of the Jangjeon Matheatical Society,vol.5,no.2,pp.23 23, 22. O((λ (n)) α P ) Δp +O((λ (n)) α )O((λ (n)) α ). (5) This poves Theoe 6.

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