 Applied Mathematics, 2011, 2, 1448-1452 doi:10.4236/am.2011.212206 Published Online December 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator Vishnu Narayan Mishra1, Huzoor H. Khan2, Kejal Khatri1 1Department of Mathematics, S. V. National Institute of Technology, Surat, India 2Department of Mathematics, Aligarh Muslim University, Aligarh, India E-mail: vishnu_narayanmishra@yahoo.co.in, huzoorkhan@yahoo.com, kejal0909@gmail.com Received May 4, 2011; revised October 25, 2011; accepted November 5, 2011 Abstract In the present paper, an attempt is made to obtain the degree of approximation of conjugate of functions (signals) belonging to the generalized weighted W(LP, ξ(t)), (p ≥ 1)-class, by using lower triangular matrix operator of conjugate series of its Fourier series. Keywords: Conjugate Fourier Series, Generalized Weighted W(LP, ξ(t))-Class, Degree of Approximation and Lower Triangular Matrix Means 1. Introduction Let f be a -periodic signal (function) and let 2π110, 2πfLL. Then the Fourier series of a function (signal) f at any point x is given by 010()cos sin 2(;),kkkkkafxakxbufx kx (1.1) with partial sums (;)nsfx—a trigonometric polynomial of degree (or order) n, of the first terms. (1)nThe conjugate series of Fourier series (1.1) of f is given by 1(cos sin )(;)kk kkbkxakx vf1kx (1.2) with partial sums (;)nsfx. If fis Lebesgue integrable and 1,(( ),)pfLiptp , then  ππ002π()cot (2)dlimcot (2)d,hhfxttt tt t exists for all x Zygmund [1, p. 131], ()fx is called the conjugate function of ().fx The matrix , in which ,nka is the element in n-th row and k-th column is usually called the matrix of T. Matrices T such that for are called lower triangular. T( ),nka,nka0,,knLet ,T( )nka be an infinite lower triangular matrix satisfying Töeplitz  conditions of regularity, i.e. ,0nnkkaM, where M is a finite constant independent of n, ,lim 0nkna, for each and kn,0lim 1nnknka . Let be an infinite series whose partial sum 0nnu(1)thk0.kknnsu The sequence-to-sequence transformation ,,00,0,1,2,nnk knnk n kkkasas n  , defines the sequence n of lower triangular matrix summability means of sequence ns generated by the sequence of coefficients ,nk The transforms n(a). are called linear means or matrix means (determined by the matrix T) of the sequence .ns An infinite series nu is said to be summable to s by lower triangular matrix T-method, if lim nn exists and is equal to s Zygmund [1, p. 74] and we write (),nsT as The summability matrix T or the sequence- to-sequence transformation nn is said to be regular, if li limnnnnmsss . A function (signal) ()fxLip, for 01, if ()() ().fxt fxt   V. N. MISHRA ET AL.1449 A function (signal) ()( , )fxLipp for 1, 01p, Fadden , if 12π0()()d ()pp,fxt fxxt  Given a positive increasing function ()t and an in- teger 1,()(( ),),pfxLiptp Khan , if 12π0()()d (()pp).fxt fxxt  In case then () ,01,tt((),)Lipt p coin- cides with the class (,)Lipp. If in p(,Lip p) class then this class reduces to .Lip For a given positive increasing function ()t, an in- teger 1,()(,( )),ppfxWLt Khan , if 12π0()()sin d(()),(0ppfx tfxxxt ). We note that, if 0) then the generalized weighted (,( )),(1pWLtp((), ).Lip t p-class coincide with the class  Also we observe that (, )((), )(,())pLipLippLiptpW Lt for 01,p1,  Mishra . The pL-norm is defined by 120()d, 1.pppffxxp The norm of a function is defined by -L:fR Rsup() :,ffxxR and the degree of approximation of a function is given by ()nEf:fR R()Min()(;) ,nnpnEffx fx in terms of n, where (;)nfx is a trigonometric poly- nomial of degree (order) n. This method of approxima- tion is called trigonometric Fourier Approximation (tfa) Mishra . Riesz-Hölder Inequality states that if p and q be non-negative extended real numbers such that 111pq. If ,pfLab and ,,qgLab then 1.fgLab, and .bpqafgf g Equality holds if and only if, for some non-zero con- stants A and B, we have ..pqAfBgae Second Mean Value theorem for integration states that if :,Gab R is a positive monotonically decreasing function and :,ab R, is an integrable function, then a number () ()d(0)()dbxaaGttt Gatt. Here (0Ga) stands for , the existence of limaGwhich follows from the conditions. Note that it is essential that the interval (a, b] contains b. A variant not having this requirement is: If :,Gab R is a monotonic (not necessarily de- creasing and positive) function and :,ab R is an integrable function, then  a number ,xab such that () ()d(0)()d(0)()d.bxaaGtttGattGbtt bx We use the following notations: ()() (),tfxtfxt ,,,0,1,nnknr nrkAaA n0, ,0cos(12)1,() 2πsin( 2)nnknkakMt tt1t—the greatest integer not exceeding of 1t. Furthermore C will denote an absolute positive con- stant, not necessarily the same at each occurrence. Through- out this paper, we take and ,0(0 ),nkakn,0 1.nAn 2. Main Result It is well known that the theory of approximations i.e., tfa, which originated from a well known theorem of Weierstrass, has become an exciting interdisciplinary field of study for the last 130 years. These approxima- tions have assumed important new dimensions due to their wide applications in signal analysis, in general and in digital signal processing  in particular, in view of the classical Shannon sampling theorem. This has motivated by various investigators such as Qureshi ([7,8]), Khan ([4,9]) Chandra , Leindler  Mishra  discussed the degree of approximation of signals (functions) belonging to ,(,),((),LipLippLiptp)and (,())pWL t-classes by using Cesàro and Nörlund means of an infinite series. Qureshi ([12,13]) have determined the degree of ()fx, conjugate of a function ()fx Lip and (,)pLip by Nörlund means of conjugate series of a Fourier series. The purpose of this paper is to determine the degree of approximation of()fx()),(tp, conjugate of a function ()(,1),pfx WL by lower triangular matrix means. We prove: xab such that Theorem 2.1. Let ,()nkTa be an infinite regular Copyright © 2011 SciRes. AM V. N. MISHRA ET AL. 1450 lo ix such twer triangular matrhat the elements ,()nka be non-negative, non-decreasing with k ≤ n. If :fR is a 2π-periodic, Lebesgue integrable and bel the gralized weighted (,()),1pWLtpRging toonene-class, then the degree of approximation of ()fx, conjugate of ()(, ()),pfxWL t by lower trian matrix means gular(;)fxnis given by 1/(nided;)()fx(1)n0,ppfx Onn  (2.1) prov ()t ollowis positive increasinctiong fun of t satis- fying the fing conditions 1π() pntψt1) 0sin d (()pptt Onξt (2.2) 1ππ() d()()ppntψt) tOnξt (2.3) and ()is decreasing in ξttt (2.4) where is an arbitrary number such that (1) 10,qδ>+ q the conjugate index ld uniformly in x and 111p+q= . Note 1. Condition (2.4) implies of p and con- ditions (2.2), (2.3) hoπ π1,nnfor π1nn Nrote 2. Also fo 0 our Theorem (2.1) f Lal mas o prove our Theorem 2.1, we requinder the conditions of our reduces to re the fol- Theorem 2.1 on. Lem order tone of the theorem oand Kushwaha . 3 Inlowing lemma. Lemma 3.1. U ,(),nka we have ,() ,nnAMtO t for ππ.tn Proof. For 1ππ,nt 1(sin )π2,for0 π2ttt, ,n we have 1,,01,,0,1010,,()t1cos(12)1cos(12)2πsin( 2)2πsin( 2)11cos( 12)cos( 12)2212maxcos(1 2)2cos(12)12nnnnk nkkknnnnk nkkknrnn rn knnkknnMkt ktaattaktaktttakttaktaOt  1,,nnAt and ,,,,1,,1 ,1,1,1,1,1 (1)11nnnkknnn nnnnnn nnnnnnnnnnAaaa aaa aaatat    Therefore, ,() nnAMtO t This completes the proof of Lemma 3.1. 4. Proof of Theorem 2.1 The partial sum of the conjugate series the Fou-rier sees (1.2) is given by thk riofπ0π0π0π01(;)cot(2)()d2π1cos(12)()d2πsin( 2)1(;)cot(2)()d2π1cos(1 2)2πsin( 2)ntt()dnnsfxt ttnttttsfxt tttt Then π,00π,001(;)cot(2)()d2π1cos(1 2)()d2πsin( 2)nnk nknnkkasfxt ttntattt or, 12(;)()nfxfxII (4.1) Using Riesz-Hölder’s inequality, condition (2.2), (2.4), note 1, the fact that 1π(sin),for 0π22ttt, 1pq11integrals, we fi and the second mean value theorem for nd Copyright © 2011 SciRes. AM V. N. MISHRA ET AL.1451 1π101π01π01π() cos( 12)qqntkt0sin stt2() sin d()() () dsin() sin d()din (2)sinppnqqnnppnttIttttMt ttttt tttttntt 1()tOO  1πqqn01π21π21π(2 )1d1π() d; 0sin(π)1()d; 01πd1qqqnhqqnhqnqhtntOOthnn ttOthntOOttnnOn n        211/ 1 qpOnOn n (4.2) Now by Riesz-Hölder’s inequality, conditions (2.3), (2.4), note 1, Lemma 3.1, the fact that 1π(sin), for0π22ttt,111pq, we obtain π2π11πππ1ππ1/π,() dsinqqnAtOtttt()1()11π121π1π21π111(1 )d(π)dπ11ππ()11qqqqnqnqqqyyOn yynyOn nynOn nqOn Onn         111 1(1)(1). (4.3)qqpOnnOnn  π,() ()d() ()( )sindd() sin( )sind()()nnpqpqnnnppnnnItMtttM ttt tttttttt ttttAOn t1π1πdqqnt Since A has non-negative entries and row sums one, Combining 1I and 2I yields 1(;)()(1).pfx fxOnn nNow, using the -norm, we get pL12π012π1/012π101(;) ()(;)()d(1 d(1 )d(1) .ppnnpppppppfx fxfx fxxOn nxOnnxOn n  This completes the proof of our Theorem 2.1. 5. Applications The following corollaries can be derived fromheo- rem 2.1. If our TCorollary 5.1. 0 and generalized weighted class () ,01,tt ,()pWLt re- then the duces to class (,Lip )p and () Lipthe tion of a functidegree of approxima- )on fx ( ,p is given by 1p(;)().npfx fxOn Proof of corollary 5.1. From our Theorem 2.1 for , we have 0p12π011(1 ), 1.(;)()(;)()d1pnnpppfxfxfxfx x  e proof of corollary 5.1. On nOpnThis completes thCorollary 5.2. If p in corollary 5.1, then for Copyright © 2011 SciRes. AM V. N. MISHRA ET AL. Copyright © 2011 SciRes. AM 1452 01, (;)().npfx fxOn Corollary 5.3. If 1,,0,()nknknnapPP tt then the degree of approximation of (),fx conjugate o(,)f fLip p by Nörlund means 1(;)nnk0(;)nnkkfxP the conjugate series psfx ofof Fourier series is given by 1(;)().pnpfx fxOn rollary 5.4.Co If then the degree 1,,0,(),nknknnapPP ttp of approximation of (),fx conjugate of fLip by Nörlmeans und 10nnn nkkkPpsries is given by  of ate series of Fourier sethe conjuglog(nf(1),0 1,On Corollary 5.5. If such that 1)π(1), 1.Onen(5.1) 1,nknkknapqR0,qRy0nkkknn yRp is monotg then the degreonic non-decreas-ine of approximation of (),fx con- jugate of a function ,fLipgeneralized Nö10)n by means rlund (; (;)nnnkkkkfxR pqsfx of the conju- gate series (1.2) satisfies equatio6. Remarks R Kushwaha . The degree oproximation n (5.1). emark 6.1. Lal andf ap- 1.pdetermined (;)()fx fxOn if valestigateoxim conjunctions be-npby Qureshs to i [13, p. 561, L. 12] tend1031 and 2p and also for otherues. Therefore, this deficiency has encouraged to invdegree of apprlonging to Lip ation of)ugate of f(,p considering 7.ents foruable sugor the improvement of this paper. of. Chris Cings, University of Sheffield, UK and AM Editorial Assi-Huang, Scientific Research Publishing,SA for their kind cooperation during communication. es Cambridge Universityte zne JournaVol. 22, . 113-119. th.org/zmath/en/journals/searc11.p AcknowledgemThe authors are grateful to his beloved parents their encouragement to his work. The authors are grateful to the referee for his valgestions and useful com- ments fThe authors are also thankful to the AM Editor in chief Pran- nstant Ms. Tian U 8. Referenc  A. Zygmund, “Trigonometric Series Vol. I,” Press, Cambridge, 1959.  O. Töeplitz, “Über Allgemeine Lineare Mittelbildun- gen,” Prace Mamatyczno—Fizyc l, 1913, pp http://www.zentralblatt-mah/?an=00003590  L. McFadden, “Absolute Nörlund Summability,” Duke Mathematical Journal, Vol. 9, 1942, pp. 168-207. doi:10.1215/S0012-7094-42-00913-X  H. H. 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