Statistical Convergence Applied to Korovkin-type ...

1 Statistical Convergence Appliedto Korovkin-type approximation TheoryOCTAVIAN AGRATINIB abes -Bolyai UniversityFaculty of Mathematics and Computer ScienceDepartment of MathematicsKog alniceanu Str., 1, 400084, present two general sequences of positive linear operators. The first is introduced by using a class ofdependent random variables, and the second is a mixture between two linear operators of discrete type. Our goal isto study their Statistical Convergence to the approximated function. This type of Convergence can replace classicalresults provided by Bohman- korovkin theorem. A particular case is Words:Positive linear operator, Bohman- korovkin theorem, Statistical Convergence , Bernstein operator,Baskakov IntroductionSince the fifties, positive linear operators (PLOs) playan important role in approximating a real valued func-tion.

2 Korovkin-type theorems furnish useful toolsin order to establish whether a sequence of PLOs isan approximation process this meaning that it con-verges strongly to the identity operator. The genuineBohman- korovkin s theorem asserts: if the positivelinear operatorsLn,n N, mapC([a,b])into it-self such that(Lnek)n 1converges toekuniformlyon[a,b],k {0,1,2}, for the test functionse0(x) = 1, e1(x) =x, e2(x) =x2,then(Lnf)n 1converges tofuniformly on[a,b]foreachf C([a,b]). HereC([a,b])is the space of allreal-valued and continuous functions defined on theinterval[a,b].Lately, new research directions targeting this areahave been developed. One of them is given by re-placing the uniform Convergence by Statistical conver-gence.

3 The remembering of this concept will be madein the next this note we focus on the presentation of somelinear positive processes and on their study in terms ofstatistical PreliminariesFollowing H. Fast [4], a sequence of real numbers(xn)n 1is said to be Statistical convergent to a realnumberL, if, for every >0, ({n N:|xn L| }) = 0,where (S) = limN 1NN j=1 S(j)is the density of the setS Nand Sstands forthe characteristic function onS. For (S)also usesthe termasymptotic density. We use the notationst limn xn= following characterization of Statistical con-vergence (cf. [6,Lemma ]) takes place. A se-quence of real numbers(xn)n 1converges statisti-cally toL Rif and only if there is a set of indicesM={nj:nj< nj+1, j N} Nwith theproperty (M) = 1andlimj xnj= main idea of Statistical Convergence of a sequenceis that the majority, in a certain sense, of its elementsconverges and we are not interested in what happensto the remaining elements.

4 Actually, the sequencesthat come from the real life sources are not conver-gent in the strictly mathematical sense. The advantageof replacing the uniform Convergence by statisticalconvergence consists in the fact that the second con-vergence is efficient in summing divergent sequenceswhich may have unbounded TRANSACTIONS on MATHEMATICSO ctavian AgratiniE-ISSN: 2224-2880183 Volume 16, 2017In approximation Theory by linear positive oper-ators, the Statistical Convergence has been examinedfor the first time in 2002 by Gadjiev and Ci-han Orhan. Bohman- korovkin criterion via statisticalconvergence will be read as 1([5,Theorem 1]).If the sequence of pos-itive linear operatorsLn:C([a,b]) B([a,b])sat-isfies the conditionsst limn Lnej ej = 0, j {0,1,2},then, for any functionf C([a,b]), we havest limn Lnf f = the aboveB([a,b])stands for the real val-ued functions bounded on the domain[a,b].

5 We getC([a,b]) B([a,b])andB([a,b])is endowed withthe uniform norm (or the sup-norm) , where f =supf B([a,b])|f(x)|.3 Two classes of operatorsWe follows closely the construction of the operatorsgiven at [2]. LetJbe given interval of the real an affine substitution maps(a,b), a <b , onto(0,1),R += (0, )orR, it is enoughto consider these intervals as beingint(J).LetIn,n N, be the sets of indices such thatIn In+1holds. We consider two , thus a model can be chosen{0,1,..,sn}or{ sn,..,0,..,sn}.Inis infinite, thus our modelcan be consideredN0={0} NorZ. For each integern 1we consider a net onJnamely(kn )k In,where >0is a fixed start from a sequence(Ln)n 1of linear posi-tive operators of discrete type given by the formula(Lnf)(x) = k Inan,k(x)f(kn ), x J,(1)wherean,k C(J)andan,k 0for every(n,k)belonging toN In.

6 HereFbelongs to a vectorialsubspace ofRJsuch that the operatorsLn,n 1, arewell defined. Regarding the above operators we re-quire the following conditions to be fulfilled for eachn NLne0=e0, Lne1=e1,(2)Lne2=e2+ n,(3)where n C(J). Operators satisfying relations (2)are called of Markov type. Further on, letXbe a nonconstant real random variable on a probability space( ,F,P). Denoting by its probability density func-tion, we assume that L2(R)and has a compactsupport included inJ. This implies L1(R). Also, being a density function, one has 0and 1= R (t)dt= setE(X) =e, V ar(X) = 2,the expectation and the variance ofX, fromXwe generate the random variablesXn,kdefined byXn,k=1n (X+k e),(n,k) N In.(4)SinceXis non-constant, by examining (4) we de-duce that for any(k1,k2) In In, the variablesXn,k1,Xn,k2are not independent.

7 All these variablesrepresent scaled versions of the same variableX, theybeing obtained from it by contractions(n , n N)and by translations((k e)n , k In).We getE(Xn,k) =kn andV ar(X) = 2n2 .(5)The expectations ofXn,k,k In, represent exactlythe mesh :={f:R R, E(|f Xn,k|)< , (n,k) N In},we introduce the operators n:S C(J),n N,as follows nf= k Inan,kE(f Xn,k) = k Inan,k f Xn,kdP,this meaning( nf)(x) =n k Inan,k(x) Rf(t) (n t k+e)dt,(6)x is easy to see that noperators are linear present the next class of operators we returnto relation (1). The operatorsLn,n 1, are fullydetermined if we specify the intervalJ, the set of in-dicesIn, the system of nodes considered and the func-tionsan,k,(n,k) N In. This time we consider aWSEAS TRANSACTIONS on MATHEMATICSO ctavian AgratiniE-ISSN: 2224-2880184 Volume 16, 2017completely arbitrary network nodes(xn,k).

8 Also, inrelation (3) we assume that n(x) = n(x)u(n), n C(J)andu(n) =O(n ), n ,for some constant >0. In short this informationwill be write as followsLn: J,In,xn,k,an,k, n,u ,(n,k) N be two sequences of this type, namelyL(1)n: [0,1],In,xn,k,a(1)n,k, 1,n,u1 ,(n,k) N In,L(2)n: [0, ),Jn,xn,k,a(2)n,k, 2,n,u2 ,(n,k) N Jn,such that0 In Jnandxn,0= 0. Regardingthese sequences we impose the following admissiblecondition to be satisfied: for anyn N, a function n C([0, ))exists such thatxn,p 2,p(x) =u2(p) n(x)u1(n), p In, x 0.(7)If this condition is fulfilled then the pair(L(1)n,L(2)n)forms a compatible couple of approximation pro-cesses, see [1].Finally we consider a function C([0, ))such that0 (x) 1for everyx 0. The an-nounced sequence of operators are defined as follows(Ln, f)(x) = p In k Jpa(1)n,p( (x))a(2)p,k(x) f(xn,pxp,k+ (1 xn,p)x),(8)x 0, wherefbelongs to a space such that the oper-ators are well particular case can be obtained by choosingL(1)n BnandL(2)n Vn, Bernstein and Baskakovoperator ofnthorder, respectively.]]]

9 We recall(Bnf)(x) =n k=0(nk)xk(1 x)n kf(kn),x [0,1],(Vnf)(x) = k=0(n+k 1k)xk(1 +x)n+kf(kn),x identifyxn,k=k/n,u1(n) =n N,u2(p) =p N0, 2,p(x) =x+x2,x 0. Tak-ing n(x) =x2+x,x 0. condition (7) is this case the operatorsLn, have been introducedand studied by F. Altomare and Mangino [3].We also mention that the operators defined by (8) arepositive and approximation propertiesThe main results will be read as 2 Let n,n N, be the operators given at(6) such thatst limn n K= 0.(9)For any functionfcontinuous on a compactK J,the following relationst limn nf f K= 0(10)takes place, where the norm Kis computed onlyfor functions restricted first step we estimate nej,j {0,1,2}. ne0= k Inan,k=Lne0=e0,(11)see (2). ne1= k Inan,kE(Xn,k)= k Inan,ke1(kn )=Lne1=e1,(12)see (5) and (2).

10 Ne2= k Inan,kE(X2n,k)= k Inan,k(V ar(Xn,k) +E2(Xn,k))= 2n2 Lne0+Lne2=e2+ n+ 2n2 ,(13)see (5) and (3).Relying on what we have achieved in the previ-ous step, we verify the accomplishment of conditionsrequired by Theorem 1. Forj= 0andj= 1, it isclear nej ej K= 0. Also, by using (13), we get ne2 e2 K= n+ 2n2 K n K+ 2n2 ,consequentlyst limn ne2 e2 K= 0. We usedrelation (9).Since the requirements of Theorem 1 are satisfied,the identity (10) takes place and the proof is TRANSACTIONS on MATHEMATICSO ctavian AgratiniE-ISSN: 2224-2880185 Volume 16, 2017 Theorem 3 LetLn, ,n N, be the operators givenat (8), such that the functions nsatisfy the property C >0,| n(x)| C, x 0, n N,whereCis independent ofn. For any functionfcon-tinuous on a compactK [0, ), the following re-lationst limn Ln, f f K= 0(14)takes using [1,Theorem 1] the following identi-ties holdLn, ej=ej, j {0,1}, Ln, e2=e2+ nu1(n).]