Iterates of a class of discrete linear operators via contraction principle

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1 Comment.Math.Univ.Carolin. 44,3(2003) Iterates of a class of discrete linear operators via contraction principle Octavian Agratini, Ioan A. Rus Abstract. In this paper we are concerned with a general class of positive linear operators ofdiscretetype.basedontheresultsoftheweaklypicardoperatorstheoryouraimisto study the convergence of the iterates of the defined operators and some approximation properties of our class as well. Some special cases in connection with binomial type operators are also revealed. Keywords: linear positive operators, contraction principle, weakly Picard operators, delta operators, operators of binomial type Classification: 41A36, 47H10 1. Introduction In the late decades the second author developed the theory of weakly Picard operators, see e.g.[9],[10],[11]. For the convenience of the reader, the basic featuresofthetheorywillbepresentedinthenextsection. Herewealsocomprised elements about delta operators and their basic polynomials. Further on, insection3weconstructageneralclassoflinearpositiveoperatorsactingonthe space C([a, b]) and we study the convergence of their iterates. Simultaneously, approximation properties of our family of operators are investigated. ThefocusofSection4istopresentconcreteexamplesofourapproach.They are in connection with approximation operators of binomial type. This way, results regarding the iterates of Bernstein, Stancu and respectively Cheney-Sharma operators are obtained and a bridge between the contraction principle and approximation of functions by binomial polynomials is built up. 2. Notation and preliminaries Definition1([9]). Let(X, d)beametricspace.theoperator A:X Xisa weaklypicardoperator(wpo)ifthesequenceofiterates(a m (x)) m 1 converges forall x Xandthelimitisafixedpointof A. Iftheoperator AisWPOand F A = {x }then AiscalledaPicardoperator(PO). Here F A := {x X A(x)=x}standsforthefixedpointsetof Aand,as usually,weput A 0 = I X, A m+1 = A A m, m N. Moreover we have the following characterization of the WPOs.

2 556 O. Agratini, I.A. Rus Theorem1([9]). Let(X, d)beametricspace. Theoperator A:X Xis WPOifandonlyifapartitionof Xexists, X= λ Λ X λ,suchthatforevery λ Λonehas (i) X λ I(A), (ii) A Xλ : X λ X λ isapicardoperator, where I(A) := {Y X A(Y) Y }representsthefamilyofallnon-empty invariant subsets of A. Furtheron,if AisWPOweconsider A X X definedby (1) A (x):= lim m Am (x), x X. Clearly,wehave A (X)=F A. Also,if AisWPO,then([11])theidentities F A m= F A, m N,holdtrue. In what follows, some elements regarding the delta operators are presented. Forany n N 0 = N {0}wedenotebyΠ n thelinearspaceofpolynomialsof degreelessorequalto nandbyπ nthesetofpolynomialsofdegree n. Weset Π:= n 0 Π n. Asequence b=(b n ) n 0, b n Π nforevery n N 0,iscalledof binomialtypeifforany(x, y) R Rthefollowingidentitiesaresatisfied (2) b n (x+y)= ( ) n b k k (x)b n k (y), n N. Anoperator T L:= {L:Π Π Llinear}whichcommuteswithallshift operators E a, a R,iscalledashift-invariantoperator andthesetofthese polynomialswillbedenotedby L s. Werecall: (E a p)(x)=p(x+a), p Π. Throughoutthepaper e n standsformonomials, e 0 =1and e n (x)=x n, n N. Definition2. Anoperator Qiscalledadeltaoperatorif Q L s and Qe 1 isa non-zeroconstant. L δ denotesthesetofalldeltaoperators. Apolynomialsequence p=(p n ) n 0 iscalledthesequenceofbasicpolynomials associatedto Qifonehas p 0 = e 0, p n (0)=0and Qp n = np n 1,forevery x R and n N. It was proved[8, Proposition 3] that every delta operator has a unique sequence of basic polynomials. We point out that the first rigorous version of the so-called umbral calculus belongs to Gian-Carlo Rota and his collaborators, see, e.g.,[8]. Among the most recent survey papers dedicated to this subject we quote[1],[5] and in what follows we gather some classical results and formulas concerning this symbolic calculus. Theorem2. (a)if p=(p n ) n 0 isabasicsequenceforsomedeltaoperator Q, thenitisasequenceofbinomialtype.reciprocally,if pisasequenceofbinomial type,thenitisabasicsequenceforsomedeltaoperator.

3 Iterates of a class of discrete linear operators via contraction principle 557 (b)let T L s and Q L δ withitsbasicsequence p=(p n ) n 0.Onehas (3) T= k 0 (Tp k )(0) Q k. k! (c)anisomorphismψexistsfrom(f,+, ),theringoftheformalpowerseries over Rfield,onto(L s,+, )suchthat Ψ(φ(t))=T, where φ(t)= k 0 a k k! tk and T= a k k! Qk. k 0 (d)anoperator P L s isadeltaoperatorifandonlyifitcorrespondsunderthe isomorphismdefinedby(3),toaformalpowerseries φ(t)suchthat φ(0)=0and φ (0) 0. (e)let Q L δ with p=(p n ) n 0 itssequenceofbasicpolynomials.let φ(d)= Qand ϕ(t)betheinverseformalpowerseriesof φ(t),where Drepresentsthe derivative operator. Then one has (4) exp(xϕ(t))= n 0 p n (x) t n, n! where ϕ(t)hastheform c 1 t+c 2 t (c 1 0). We accompany this brief exposition with the following Examples. Thesymbol IstandsfortheidentityoperatoronthespaceΠ. 1.Thederivativeoperator Dhasitsbasicsequencegivenby(e n ) n 0. 2.Theforwarddifferenceoperator h := E h Ihasitsbasicsequence(x [n,h] ) n 0 where x [0,h] :=1and x [n,h] := x(x h)...(x n 1h)representthegeneralized factorialpowerwiththestep h. Analogously,((x+n 1h) [n,h] ) n 0 isthesequenceofbasicpolynomialsassociatedto h := I E h,thebackwarddifference operator.itisevidentthat h = h E h. 3. Abeloperator A a := DE a, a 0,isalsoadeltaoperator. Forevery p Π, (A a p)(x)= dp dx (x+a)or,symbolically,wealsocanwrite A a= D(e ad ).TheAbel sequenceofpolynomials ã=(a n a ) n 0,where a a 0 :=1, a n a (x):= x(x na) n 1, n N,formsthesequenceofbasicpolynomialsassociatedto A a. It is not complicate to prove that all the above polynomial sequences verify relation(2).

4 558 O. Agratini, I.A. Rus 3. Asequenceofoperatorsinstudy At first we construct an approximation process of discrete type acting on the space C([a, b])endowedwiththechebyshevnorm.foreachinteger n 1 we consider the following. (i)aneton[a, b]named n isfixed(a=x n,0 < x n,1 < < x n,n = b). (ii)asystem(ψ n,k ),n isgiven,whereevery ψ n,k belongsto C([a, b]). We assumethatitisablendingsystemwithacertainconnectionwith n,more precisely the following conditions hold: (5) ψ n,k 0(, n), Now we define the operators ψ n,k = e 0, x n,k ψ n,k = e 1. (6) L n : C([a, b]) C([a, b]), (L n f)(x)= ψ n,k (x)f(x n,k ). Remark1. L n, n N,arepositivelinearoperatorsandconsequentlytheybecomemonotone.Takingintoaccount(5)wehave L n e 0 = e 0, L n e 1 = e 1. Moreover L n :=sup f 1 L nf =1,forevery n N.Weindicatethenecessary andsufficientconditionwhichoffersto(l n ) n 1 theattributeofapproximation process. Theorem3. Let L n, n N,bedefinedby(6). (i) If lim n nk=1 x 2 n,k ψ n,k= e 2 uniformlyon[a, b]thenforevery f C([a, b])onehaslim n L n f= funiformlyon[a, b]. (ii) Forevery f C([a, b]), x [a, b]and δ >0,onehas (L n f)(x) f(x) (1+δ 1( n x 2 n,k ψ n,k(x) x 2) 1/2 ) ω 1 (f, δ), where ω 1 (f, )representsthemodulusofcontinuityof f. Proof: The first statement results directly from the theorem of Bohman-Korovkinandrelations(5)aswell. Thesecondstatementholdstruebyvirtueofthe classical results regarding the rate of convergence, see e.g. the monograph[2, Theorem 5.1.2]. Our main objective is to study the convergence of the iterates of our operators. Westateandprove

5 Iterates of a class of discrete linear operators via contraction principle 559 Theorem4. Let L n, n N,bedefinedby(6)suchthat ψ n,0 (a)=ψ n,n (b)=1. Letusdenote u n :=min x [a,b] (φ n,0 (x)+φ n,n (x)). If u n >0thentheiteratessequence(L m n) m 1 verifies (7) lim m (Lm n uniformly on[a, b]. Proof:Atfirstwedefine f(b) f(a) f)(x)=f(a)+ (x a), f C([a, b]), b a X α,β := {f C([a, b]) f(a)=α, f(b)=β}, (α, β) R R. Clearly,every X α,β isaclosedsubsetof C([a, b])andthesystem X α,β,(α, β) R R,makesupapartitionofthisspace. Since ψ n,0 (a)=1and ψ n,n (b)=1,therelations(5)imply(l n f)(a)=f(a) and(l n f)(b)=f(b),inotherwordsforall(α, β) R Rand n N, X α,β isan invariantsubsetof L n. Furtheron,weprovethat L n Xα,β : X α,β X α,β isacontractionforevery (α, β) R Rand n N. Indeed,if f and gbelongto X α,β then,forevery x [a, b],wecanwrite n 1 (L n f)(x) (L n g)(x) = ψ n,k (x)(f g)(x n,k ) k=1 n 1 ψ n,k (x) f g k=1 =(1 φ n,0 (x) φ n,n (x)) f g (1 u n ) f g, andconsequently, L n f L n g (1 u n ) f g.theassumption u n >0 guarantees our statement. Ontheotherhand,thefunction p α,β := α+((β α)/(b a))(e 1 a)belongs to X α,β andsince L n reproducestheaffinefunctions, p α,β isafixedpointof L n. Forany f C([a, b])onehas f X f(a),f(b) and,byusingthecontraction principle,wegetlim m L m n f= p f(a),f(b). We obtained the desired result(7).

6 560 O. Agratini, I.A. Rus Remark 2. Following the lines of the familiar Bohman-Korovkin arguments we indicate a necessary and sufficient condition for the iterates of our sequence (L n ) n 1 toconvergetotheidentityoperator.considering(k n ) n 1 anincreasing sequence of positive integer numbers tending to infinity, we enunciate: lim n L kn n f f =0foreach f C([a, b])ifandonlyifthesamelimit relationholdsfor f= e 2. TakingtheadvantageofDefinition1,Theorem1andrelation(1)aswell,for X= C([a, b])weobtain Corollary. UnderthehypothesisofTheorem4,theoperator L n iswpofor every n Nand L n f= c 1(f)e 1 + c 2 (f), f C([a, b]), where c 1 (f):=(f(b) f(a))/(b a)and c 2 (f):=(bf(a) af(b))/(b a). Actually,(L n ) n 1 isawideclassofdiscreteoperatorsandinthenextsection we show that it includes the so-called binomial operators. 4. Application KeepinginmindthedatumofSection2,let Qbeadeltaoperatorand p= (p n ) n 0 beitssequenceofbasicpolynomialsundertheadditionalassumption that p n (1) 0forevery n N.Also,accordingtoTheorem2weshallkeepthe samemeaningofthefunctions φand ϕ.forevery n 1weconsidertheoperator L Q n: C([0,1]) C([0,1])definedasfollows: (8) (L Q n f)(x)= 1 p n (1) ( ) n p k k (x)p n k (1 x)f ( ) k, n N. n They are called(cf. e.g., P. Sablonniere[12]) Bernstein-Sheffer operators, but as D.D. Stancu and M.R. Occorsio motivated in[14], these operators can be named Popoviciu operators. In 1931 Tiberiu Popoviciu[7] indicated the construction(8), infrontofthesumappearingthefactor d 1 n fromtheidentities (1+d 1 t+d 2 t ) x =exp(xϕ(t))= p n (x)t n /n!, see(4).ifwechoose x=1itbecomesobviousthat d n = p n (1)/n!.Inwhatfollows weprovethattheoperators L Q n, n N,areparticularcasesoftheoperators definedby(6).firstly,in(6)wechoose a=0, b=1,and (9) x n,k = k ( ) n n, ψ n,k(x)= p k k (x)p n k (1 x)/p n (1),, n. n=0

7 Iterates of a class of discrete linear operators via contraction principle 561 Choosingin(2) y:=1 x,from(8)weobtain L Q n e 0 = e 0. Thepositivityoftheseoperatorsaregivenbythesignofthecoefficientsofthe series ϕ(t)=c 1 + c 2 t+... (c 1 0). Moreprecisely,T.Popoviciu[7]andlater P. Sablonniere[12, Theorem 1] have established that Lemma1. L Q n isapositiveoperatoron C([0,1])forevery n 1ifandonlyif c 1 >0and c n 0forall n 2. Amongthemostsignificantresultsconcerning L Q n operators(undertheassumptions of Lemma 1) we recall: (i) (10) L Q n e 1= e 1, n 1, and L Q n e 2= e 2 + a n (e 1 e 2 ), n 2, where na n =1+(n 1)r n 2 (1)/p n (1),thesequence(r n (x)) n 0 beinggenerated by ϕ (t)exp(xϕ(t))= n 0r n (x)t n /n!; (ii) L Q n fconvergesuniformlyto f C([0,1])ifandonlyifthecondition holds. These results allow us to state lim (r n 2(1)/p n (1))=0 n Lemma2. Let L Q n, n N,bedefinedby(8).UnderthehypothesisofLemma1 these operators satisfy the requirements(5) and consequently they are particular casesof L n definedonthespace C([0,1])byformula(6). Sincethebasicpolynomial p 0 satisfies p 0 = e 0,from(9)weget ψ n,0 (0)= ψ n,n (1)=1.ExaminingTheorem4weeasilydeduce Theorem5. Let L Q n, n N,bedefinedby(8)suchthattheassumptionsof Lemma1arefulfilled.Letusconsiderthepolynomials q n, n N,where q n (x):= p n (1 x)+p n (x).ifthepolynomials q n, n N,havenozeroson[0,1]thenthe iteratessequence((l Q n) m ) m 1 verifies (11) lim m (LQ n) m f= f(0)+(f(1) f(0))e 1, uniformly on[0, 1]. At the end, choosing concrete delta operators Q we reobtain some classical linear positive operators. Practically we come back to Examples given in Section 2 andweapplytheorem5.

8 562 O. Agratini, I.A. Rus 1. If Q:= D,then L D n becomesthebernsteinoperator B n, (12) (L D n f)(x) (B nf)(x)= ( ) n x k (1 x) n k f k ( ) k, x [0,1]. n Wehave q n (x)=(1 x) n + x n 1/2 n 1, x [0,1],andtheidentity(11) holdstrue,inaccordancewitharesultduetor.p.keliskyandt.j.rivlin[4, Equation(2.4)]. 2. If Q:= 1 α α, α >0,then L α 1 α n becomesthestancuoperator[13] P [α] n, (L α 1 α n f)(x) (P n [α] f)(x)= w n,k (x; α)f ( ) k, x [0,1], n where w n,k (x; α)= ( n k ) x [k, α] (1 x) [n k, α] /1 [n, α], αbeingapositiveparameterwhichmaydependonlyonthenaturalnumber n. Fortheabove Qthe basicpolynomialsaregivenby p n (x)=(x+(n 1)α) [n,α] andconsequently, q n (x) (1 x) n + x n 1/2 n 1, x [0,1]. Oncemore,ouridentity(11)harmonizes with a result established in 1978 by G. Mastroianni and Mario Rosario Occorsio[6]. 3. Wechoose Q:= A a (Abeloperator)withitsbasicsequence ã. Assuming thattheparameter aisnonpositiveanddependson n, a:= t n,oneobtains the Cheney-Sharma operator named G n, see[3]orthe monograph[2, Equation(5.3.16)]. Itisknown: if thesequence(nt n ) n 1 convergestozerothen lim n G n f f =0forevery f C([0,1]). Thistimewehave p 0 (x)=1 and p n (x)=x(x+nt n ) n 1, n N. Thus,thepolynomials q n, n N 0,haveno zeroson[0,1]and(11)holdstrue. References [1] Agratini O., Binomial polynomials and their applications in Approximation Theory, Conferenze del Seminario di Matematica dell Universita di Bari 281, Roma, 2001, pp [2] Altomare F., Campiti M., Korovkin-Type Approximation Theory and its Applications, de Gruyter Series Studies in Mathematics, Vol.17, Walter de Gruyter, Berlin-New York, [3] Cheney E.W., Sharma A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma(2) 5(1964), [4] Kelisky R.P., Rivlin T.J., Iterates of Bernstein polynomials, Pacific J. Math. 21(1967), [5] Lupaş A., Approximation operators of binomial type, New developments in approximation theory(dortmund, 1998), pp , International Series of Numerical Mathematics, Vol.132, Birkhäuser Verlag Basel/Switzerland, [6] Mastroianni G., Occorsio M.R., Una generalizzatione dell operatore di Stancu, Rend. Accad. Sci. Fis. Mat. Napoli(4) 45(1978),

9 Iterates of a class of discrete linear operators via contraction principle 563 [7] Popoviciu T., Remarques sur les polynômes binomiaux, Bul. Soc. Sci. Cluj(Roumanie) 6 (1931), (also reproduced in Mathematica(Cluj) 6(1932), 8 10). [8] Rota G.-C., Kahaner D., Odlyzko A., On the Foundations of Combinatorial Theory. VIII. Finite operator calculus, J. Math. Anal. Appl. 42(1973), [9] Rus I.A., Weakly Picard mappings, Comment. Math. Univ. Carolinae 34(1993), no. 4, [10] Rus I.A., Picard operators and applications, Seminar on Fixed Point Theory, Babeş-Bolyai Univ., Cluj-Napoca, [11] Rus I.A., Generalized Contractions and Applications, University Press, Cluj-Napoca, [12] Sablonniere P., Positive Bernstein-Sheffer operators, J. Approx. Theory 83(1995), [13] Stancu D.D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 13(1968), no. 8, [14] Stancu D.D., Occorsio M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numér. Théor. Approx. 27(1998), no. 1, Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. Kogălniceanu 1, 3400 Cluj-Napoca, Romania agratini@math.ubbcluj.ro iarus@math.ubbcluj.ro (Received March 11, 2002)