JFSAJournal of Function Spaces and Applications0972-68022090-8997Hindawi Publishing Corporation17831610.1155/2012/178316178316Research ArticleIdeal Convergence of k-Positive Linear OperatorsGadjievAkif1DumanOktay2GhorbanalizadehA. M.1CarroMaria1Institute of Mathematics and MechanicsNational Academy of Sciences of Azerbaijan9 F. Agaev Street1141 BakuAzerbaijanscience.az2Department of MathematicsFaculty of Arts and SciencesTOBB University of Economics and TechnologySöğütözü, 06530 AnkaraTurkeyetu.edu.tr2012412012201203032010170620102012Copyright © 2012 Akif Gadjiev et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study some ideal convergence results of k-positive linear operators defined on an appropriate subspace of the space of all analytic functions on a bounded simply connected domain in the complex plane. We also show that our approximation results with respect to ideal convergence are more general than the classical ones.

1. Introduction

The classical Korovkin theory (see ) is mainly based on the “positivity” of real-valued linear operators. In , this theory was improved by Gadjiev and Orhan via the concept of statistical convergence (see, also, ). However, in order to obtain the Korovkin-type results for complex-valued linear operators, the concept of “k-positivity” introduced by Gadžiev  is used instead of the classical positivity. Such results may be found in the papers . In the recent works [11, 12], with the help of some convergence methods, such as A-statistical convergence and ideal convergence, various approximation theorems have been obtained for k-positive linear operators defined on some appropriate subspaces of all analytic functions on the unit disk. In the present paper, we study some ideal convergence results of a sequence of k-positive linear operators on a bounded simply connected domain that does not need to be the unit disk. Furthermore, we present a general family of k-positive linear operators which satisfy all conditions of our results but not the classical ones.

We first recall the concept of “ideal convergence.

Let X be a nonempty set. A class of subsets of X is said to be an ideal in X provided that

ϕ (the empty set) ,

if A,B, then AB,

if A and BA, then B.

An ideal is called nontrivial if X. Also, a nontrivial ideal in X is called admissible if {x} for each xX (see  for details). In [14, 15], using the above definition of ideal, a new convergence method which is more general than the usual convergence has been introduced as follows.

Let be a nontrivial ideal in , the set of all positive integers. A sequence x={xn}n is ideal convergent (or -convergent) to a number L if, for every ε>0, {n:|xn-L|ε}, which is denoted by -limnxn=L. Notice that the method of ideal convergence includes many convergence methods such as A-statistical convergence, statistical convergence, lacunary statistical convergence, usual convergence, and so forth. For example, if is the class of all finite subsets of , then -convergence reduces to the usual. Furthermore, -convergence coincides with the concept of A-statistical convergence (see ) by taking ={K:δA(K)=0}, where A is a nonnegative regular summability matrix and δA(K) denotes the A density of K. Besides, if we choose A=C1, the Cesáro matrix of order one, then we immediately obtain the statistical convergence (see ). With these properties, using the ideal convergence in the approximation theory provides us many advantages.

Now, we also recall some basic definitions and notations used in the paper.

Let D be a bounded simply connected domain in , the set of all complex numbers. By A(D) we denote the space of all analytic functions on D. Let ϕ(z) be any analytic function mapping D conformally and one to one on the unit disk. Then, for every fA(D), we have the Taylor expansion of f given by f(z)=k=0fk(ϕ(z))k, where fk, k0={0}, is the Taylor coefficient of f satisfying limsupk|fk|k1. It is known that Taylor’s coefficients are calculated by the following formula (see ): fk=12πiCf(z)ϕ(z)(ϕ(z))k+1dz, where C is any closed contour lying in the interior of D. It is not hard to see that the series (1.1) under the condition (1.2) is uniformly convergent if |ϕ(z)|=r<1. It is well known that A(D) is Fréchet’s space with topology of compact convergence in any closed subset of D. In this paper, we use the norm ·r on the space A(D) defined by fr:=max|ϕ(z)|=r<1|f(z)|,for  fA(D), and therefore the convergence in A(D) is the convergence in norm ·r for any 0<r<1.

Now let A(D+):={fA(D):fk0,k=0,1,2,}. Following  (see also ), if a linear operator T mapping A(D) into itself satisfies the condition T(A(D+))A(D+), then we say that T is a “k-positive linear operator”. As usual, for a function fA(D), the value of T(f) at a point z is denoted by T(f;z) and also the Taylor expansion of T(f) is T(f;z)=k=0(m=0Tk,mfm)(ϕ(z))k, where fm(m0) is the Taylor coefficient of f and Tk,m(k,m0) is the Taylor coefficient of T([ϕ(z)]k). If {Tn}n is a sequence of k-positive linear operators from A(D) into itself, then, as in (1.6), we may write that Tn(f;z)=k=0(m=0Tk,m(n)fm)(ϕ(z))k,for  each  nN, where, of course, m=0Tk,m(n)fm is the Taylor coefficient of Tn(f) for n and k0. Observe that Tn is k-positive if and only if Tk,m(n)0 for every n and k,m0.

Throughout the paper we use the following three test functions: ei(z):=k=0ki(ϕ(z))k,i=0,1,2.

We also consider the following subspace of A(D): A*(D):={fA(D):|fk|M(1+k2)  for  every  kN0  and  for  some  M>0}.

2. Ideal Convergence of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M102"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math></inline-formula>-Positive Linear Operators

In order to compute the degree of ideal convergence of sequences we introduce the following definition (see also ).

Definition 2.1.

Let be an admissible ideal in . Then, one says that a sequence {xn}n is ideal convergent to a number L with degree 0<β1 if, for every ε>0, {nN:|xn-L|n1-βε}I. In this case, we write xn-L=I-o(n-β),as  n.

Notice that if we choose, in particular, β=1 in Definition 2.1, we immediately get the ideal convergence of (xn) to L. Observe that, according to Definition 2.1, the degree β is controlled by the entries of the sequence {xn}n.

We first need the following two lemmas.

Lemma 2.2.

Let be an admissible ideal in . Assume further that {fn}n is a sequence of analytic functions on D with the Taylor coefficients fn,k for each n and k0. Then, for 0<β1, fnr=I-o(n-β),as  n, if and only if there exists a sequence {tn,k}n,k0 for which the following conditions hold: |fn,k|tn,k,for  every  nN,  kN0,limsupktn,kk1,for  each  fixed  nN,tn,k=I-o(n-β),as  n,for  each  fixed  kN0.

Proof.

Necessity.

Assume that (2.3) holds. Then, choosing tn,k=|fn,k|, the conditions (2.4) and (2.5) can be obtained immediately. Now we prove (2.6). By (2.3), we may write that, for each fixed k0, εn:=fnr=max|ϕ(z)|=r<1|fn(z)|=I-o(n-β),as  n. By (1.3), for any r<1, we obtain that tn,k=|fn,k|12π|ϕ(z)|=r|fn(z)||ϕ(z)||ϕ(z)|k+1|dz|εnrk,for  nN,kN0. Hence, it follows from (2.8) that, for a given ε>0, {nN:tn,kn1-βε}{nN:εnn1-βrkε}. By (2.7), we get {nN:εnn1-βε}I, where ε=rkε. Therefore, from (2.9), we conclude that {nN:tn,kn1-βε}I, which gives (2.6).

Sufficiency.

Assume now that conditions (2.4)–(2.6) hold. It follows from (2.5) that the series k=0tn,krk is convergent for any r<1. Then, for every ε>0, there exists a positive natural number N=N(ε) such that k=N+1tn,krε holds. Using (2.4) and (2.12) and also considering the fact that fnrk=0|fn,k|rkk=0Ntn,krk+k=N+1tn,krk, we get fnrε+k=0Ntn,krk, which yields that fnrn1-βεn1-β+k=0Ntn,krkn1-β. Since rk1 for every k=0,1,,N and 1/n1-β1 for every n and 0<β1, we obtain that fnrn1-βε+k=0Ntn,kn1-β. Now, for a given γ>0, choose an ε>0 such that ε<γ. Then, define the following sets: E:={nN:fnrn1-βγ},Ek:={nN:tn,kn1-βγ-εN+1},k=0,1,,N. From (2.16), it is clear that Ek=0NEk. By hypothesis (2.6), we know Ek for each k=0,1,,N. Hence, by the definition of ideal, we immediately obtain that the set E belongs to for every γ>0, which implies (2.3). So, the proof is completed.

Lemma 2.3.

Let be an admissible ideal in , and let {Tn}n be a sequence of k-positive linear operators from A(D) into itself. If, for some βi with 0<βi1(i=0,1,2), Tn(ei)-eir=I-o(n-βi),as  n, holds, then there exists a sequence {tn,k}n,k0 satisfying (2.5), (2.6) with β:=min{β0,β1,β2} such that the following inequality m=0Tk,m(n)(m-k)2(k+1)2tn,k holds, where Tk,m(n) is the same as (1.7).

Proof.

We first observe that m=0Tk,m(n)(m-k)2=(m=0Tk,m(n)m2-k2)-2k(m=0Tk,m(n)m-k)+k2(m=0Tk,m(n)-1)|m=0Tk,m(n)m2-k2|+2k|m=0Tk,m(n)m-k|+k2|m=0Tk,m(n)-1|. By (2.19) and Lemma 2.2, there exist sequences {ti,n,k}n,k0 and numbers βi with 0<βi1(i=0,1,2) satisfying (2.5), (2.6) such that the following conditions |m=0Tk,m(n)-1|t0,n,k,|m=0Tk,m(n)m-k|t1,n,k,|m=0Tk,m(n)m2-k2|t2,n,k hold. Hence, we easily get m=0Tk,m(n)(m-k)2(k+1)2tn,k, where tn,k:=max{t0,n,k,t1,n,k,t2,n,k}. Since β:=min{β0,β1,β2}, we can see that tn,k=o(n-β),as  n, whence the result.

Then, we are ready to give our main result.

Theorem 2.4.

Let be an admissible ideal in , and let {Tn}n be a sequence of k-positive linear operators from A(D) into itself. If, for some βi with 0<βi1(i=0,1,2), (2.19) holds, then, for every fA*(D), one has Tn(f)-fr=I-o(n1-β),as  n, where β:=min{β0,β1,β2}.

Proof.

Assume that (2.19) holds for some βi with 0<βi1(i=0,1,2). Let fA*(D) and zD be fixed. By (1.1) and (1.7), we may write that Tn(f;z)-f(z)=k=0(ϕ(z))km=0Tk,m(n)(fm-fk)+k=0(ϕ(z))kfk(m=0Tk,m(n)-1). The last inequality gives that Tn(f)-frk=0rkm=0Tk,m(n)|fm-fk|+k=0rk|fk||m=0Tk,m(n)-1|. By Lemma  2.2(ii) of , we know the fact that |fm-fk|M(3+k)4(m-k)2. Also, since fA*(D), we see that |fk|M(1+k2). Hence, combining these inequalities, we have Tn(f)-frMk=0rk(3+k)4m=0Tk,m(n)(m-k)2+Mk=0rk(1+k2)|m=0Tk,m(n)-1|. By Lemma 2.3, there exist a sequence {tn,k}n,k0 satisfying (2.5), (2.6) with β:=min{β0,β1,β2} such that the inequality Tn(f)-frM(k=0rk(3+k)4(k+1)2tn,k+k=0rk(1+k2)(k+1)2tn,k) holds. Hence, we get, for every n, that Tn(f)-frMk=0rk(3+k)6tn,k. Since, for every k,n, rk(3+k)6tn,krk46k6tn,k, we easily get that {rk(3+k)6tn,k}1/kr46/kk6/ktn,k1/k. Now taking limsup as k in both sides of the last inequality and also using (2.5), we observe that the series in the right-hand side of (2.30) converges for any 0<r<1. Therefore, the remain of the proof is very similar to the proof of the sufficiency part of Lemma 2.2.

3. Concluding Remarks

In this section, we give some useful consequences of our Theorem 2.4.

We first observe that, for |ϕ(z)|=r<1 with ϕ(z)0, 11-ϕ(z)=k=0(ϕ(z))k=e0(z),ϕ(z)(1-ϕ(z))2=ϕ(z)ϕ(z)ddz(11-ϕ(z))=ϕ(z)k=0k(ϕ(z))k-1=k=0k(ϕ(z))k=e1(z),ϕ2(z)(1-ϕ(z))3=ϕ2(z)2ϕ(z)ddz(1(1-ϕ(z))2)=ϕ2(z)2k=0k(k-1)(ϕ(z))k-2=12k=0k(k-1)(ϕ(z))k=e2(z)-e1(z)2. Then, defining fi(z):=(ϕ(z))i(1-ϕ(z))i+1,i=0,1,2, the next result is equivalent to Theorem 2.4.

Theorem 3.1.

Let be an admissible ideal in , and let {Tn}n be a sequence of k-positive linear operators from A(D) into itself. If, for some βi with 0<βi1(i=0,1,2), Tn(fi)-fir=I-o(n-βi),as  n, then, for every fA*(D), (2.25) holds for the same β as in Theorem 2.4.

If we take β=1 in Theorems 2.4 and 3.1, then we immediately get the following characterization for ideal approximation by k-positive linear operators.

Corollary 3.2.

Let be an admissible ideal in , and let {Tn}n be a sequence of k-positive linear operators from A(D) into itself. Then, for every fA*(D), I-limnTn(f)-fr=0, if and only if, for each i=0,1,2, I-limnTn(ei)-eir=0, or equivalently, I-limnTn(fi)-fir=0,

Proof.

Since each ei,fiA*(D), the implications (3.4) (3.5) and (3.4) (3.6) are obvious. The sufficiency immediately follows from Definition 2.1 and Theorems 2.4 and 3.1.

Finally, if we choose ={K:δA(K)=0}, where A is a nonnegative regular summability matrix and δA(K) denotes the A density of K, and also if we take D={z:|z|<1}, then from Theorem 2.4 we obtain a slight modification of the result proved in .

It is known from  that if we choose ={K:δ(K)=0}, where δ(K) denotes the asymptotic density of K given by δ(K):=limn#{kn:kK}n  (provided  the  limit  exists), then -convergence reduces to the concept of statistical convergence which was first introduced by Fast . In the last equality, by #{B} we denote the cardinality of the set B. Hence, let {un}n be a sequence whose terms are defined by un:={nn+1,if  n=m2,mN,  0,otherwise. Then, we easily observe that I-limnun=st-limnun=0 holds. Assume now that {Tn}n is any sequence of k-positive linear operators from A(D) into itself, such that, for every fA*(D), the sequence {Tn}n is uniformly convergent to f on a bounded simply domain D with respect to any norm ·r(0<r<1). Then, consider the following operators: Ln(f;z):=(1+un)Tn(f,z). Therefore, observe that {Ln}n is a sequence of k-positive linear operators from A(D) into itself. By (3.10), we can write, for each i=0,1,2, that Ln(fi)-firTn(fi)-fir+unTn(fi)r(1+un)Tn(fi)-fir+un(fi)r(1+un)Tn(fi)-fir+unri(1-r)i+1. So, the last inequality gives st-limnLn(fi)-fi{st-limn(1+un)}{limnTn(ei)-ei}+st-limnunri(1-r)i+1 by our conditions on {Tn}n. Then, it follows from (3.9) that, for each i=0,1,2, st-limnLn(fi)-fi=0. Hence, by Corollary 3.2, we obtain, for every fA*(D), that I-limnLn(f)-f=st-limnLn(f)-f=0. However, by the definition (3.8), we see that, for every fA*(D), unTn(f)r={nn+1Tn(f)r,if  n=m2,mN,  0,otherwise. Now since limnTn(f)r=fr, we immediately obtain that the subsequence {unTn(f)r}n=m2 converges to fr while the subsequence {unTn(f)r}nm2 converges to zero. Hence, the sequence {unTn(f)r}n is nonconvergent. Therefore, we see that the sequence {Ln(f)}n cannot be uniformly convergent to f on D. Therefore, we can say that our ideal approximations by k-positive linear operators presented in this paper are more general and applicable than the classical ones.

Finally, for a given m, we consider the following subspace of A(D): Am*(D):={fA(D):|fk|M(1+k2m)  for  every  kN0  and  for  some  M>0}. In this case, we consider the following test functions: gi(z)=k=0kmi(ϕ(z))k,i=0,1,2. Thus, considering the same methods used in this paper, one can immediately get the following ideal approximation result on the subspace Am*(D), m.

Corollary 3.3.

Let be an admissible ideal in , and let {Tn}n be a sequence of k-positive linear operators from A(D) into itself. Then, for every fAm*(D), m, I-limnTn(f)-fr=0, if and only if, for each i=0,1,2, I-limnTn(gi)-gir=0.

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