#### Abstract

We study some ideal convergence results of -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 [1]) is mainly based on the β*positivity*β of real-valued linear operators. In [2], this theory was improved by Gadjiev and Orhan via the concept of statistical convergence (see, also, [3β6]). However, in order to obtain the Korovkin-type results for complex-valued linear operators, the concept of β*-positivity*β introduced by GadΕΎiev [7] is used instead of the classical positivity. Such results may be found in the papers [7β10]. In the recent works [11, 12], with the help of some convergence methods, such as -statistical convergence and ideal convergence, various approximation theorems have been obtained for -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 -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 -positive linear operators which satisfy all conditions of our results but not the classical ones.

We first recall the concept of β*ideal convergence.*β

Let be a nonempty set. A class of subsets of is said to be an *ideal* in provided that(i) (the empty set) ,(ii)if , then ,(iii)if and , then .

An ideal is called *nontrivial* if . Also, a nontrivial ideal in is called *admissible* if for each (see [13] 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 is *ideal convergent* (or *-convergent*) to a number if, for every , , which is denoted by . Notice that the method of ideal convergence includes many convergence methods such as -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 -statistical convergence (see [16]) by taking , where is a nonnegative regular summability matrix and denotes the density of . Besides, if we choose , the CesΓ‘ro matrix of order one, then we immediately obtain the statistical convergence (see [17]). 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 be a bounded simply connected domain in , the set of all complex numbers. By we denote the space of all analytic functions on . Let be any analytic function mapping conformally and one to one on the unit disk. Then, for every , we have the Taylor expansion of given by where , , is the Taylor coefficient of satisfying It is known that Taylorβs coefficients are calculated by the following formula (see [18]): where is any closed contour lying in the interior of . It is not hard to see that the series (1.1) under the condition (1.2) is uniformly convergent if . It is well known that is FrΓ©chetβs space with topology of compact convergence in any closed subset of . In this paper, we use the norm on the space defined by and therefore the convergence in is the convergence in norm for any .

Now let . Following [7] (see also [9]), if a linear operator mapping into itself satisfies the condition
then we say that is a β*-positive linear operator*β. As usual, for a function , the value of at a point is denoted by and also the Taylor expansion of is
where is the Taylor coefficient of and is the Taylor coefficient of . If is a sequence of -positive linear operators from into itself, then, as in (1.6), we may write that
where, of course, is the Taylor coefficient of for and . Observe that is -positive if and only if for every and .

Throughout the paper we use the following three test functions:

We also consider the following subspace of :

#### 2. Ideal Convergence of -Positive Linear Operators

In order to compute the degree of ideal convergence of sequences we introduce the following definition (see also [2]).

*Definition 2.1. *Let be an admissible ideal in . Then, one says that a sequence is ideal convergent to a number with degree if, for every ,
In this case, we write

Notice that if we choose, in particular, in Definition 2.1, we immediately get the ideal convergence of to . Observe that, according to Definition 2.1, the degree is controlled by the entries of the sequence .

We first need the following two lemmas.

Lemma 2.2. *Let be an admissible ideal in . Assume further that is a sequence of analytic functions on with the Taylor coefficients for each and . Then, for ,
**
if and only if there exists a sequence for which the following conditions hold:
*

*Proof. **Necessity. *Assume that (2.3) holds. Then, choosing , 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 ,
By (1.3), for any , we obtain that
Hence, it follows from (2.8) that, for a given ,
By (2.7), we get
where . Therefore, from (2.9), we conclude that
which gives (2.6).*Sufficiency. *Assume now that conditions (2.4)β(2.6) hold. It follows from (2.5) that the series is convergent for any . Then, for every , there exists a positive natural number such that
holds. Using (2.4) and (2.12) and also considering the fact that
we get
which yields that
Since for every and for every and , we obtain that
Now, for a given , choose an such that . Then, define the following sets:
From (2.16), it is clear that
By hypothesis (2.6), we know for each . Hence, by the definition of ideal, we immediately obtain that the set belongs to for every , which implies (2.3). So, the proof is completed.

Lemma 2.3. *Let be an admissible ideal in , and let be a sequence of -positive linear operators from into itself. If, for some with ,
**
holds, then there exists a sequence satisfying (2.5), (2.6) with such that the following inequality
**
holds, where is the same as (1.7).*

*Proof. *We first observe that
By (2.19) and Lemma 2.2, there exist sequences and numbers with satisfying (2.5), (2.6) such that the following conditions
hold. Hence, we easily get
where . Since , we can see that
whence the result.

Then, we are ready to give our main result.

Theorem 2.4. *Let be an admissible ideal in , and let be a sequence of -positive linear operators from into itself. If, for some with , (2.19) holds, then, for every , one has
**
where .*

*Proof. *Assume that (2.19) holds for some with . Let and be fixed. By (1.1) and (1.7), we may write that
The last inequality gives that
By Lemmaββ2.2(ii) of [11], we know the fact that . Also, since , we see that . Hence, combining these inequalities, we have
By Lemma 2.3, there exist a sequence satisfying (2.5), (2.6) with such that the inequality
holds. Hence, we get, for every , that
Since, for every ,
we easily get that
Now taking as 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 . 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 with , Then, defining the next result is equivalent to Theorem 2.4.

Theorem 3.1. *Let be an admissible ideal in , and let be a sequence of -positive linear operators from into itself. If, for some with ,
**
then, for every , (2.25) holds for the same as in Theorem 2.4.*

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

Corollary 3.2. *Let be an admissible ideal in , and let be a sequence of -positive linear operators from into itself. Then, for every ,
**
if and only if, for each ,
**
or equivalently,
*

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

Finally, if we choose , where is a nonnegative regular summability matrix and denotes the density of , and also if we take , then from Theorem 2.4 we obtain a slight modification of the result proved in [11].

It is known from [14] that if we choose , where denotes the asymptotic density of given by then -convergence reduces to the concept of statistical convergence which was first introduced by Fast [17]. In the last equality, by we denote the cardinality of the set . Hence, let be a sequence whose terms are defined by Then, we easily observe that holds. Assume now that is any sequence of -positive linear operators from into itself, such that, for every , the sequence is uniformly convergent to on a bounded simply domain with respect to any norm . Then, consider the following operators: Therefore, observe that is a sequence of -positive linear operators from into itself. By (3.10), we can write, for each , that So, the last inequality gives by our conditions on . Then, it follows from (3.9) that, for each , Hence, by Corollary 3.2, we obtain, for every , that However, by the definition (3.8), we see that, for every , Now since , we immediately obtain that the subsequence converges to while the subsequence converges to zero. Hence, the sequence is nonconvergent. Therefore, we see that the sequence cannot be uniformly convergent to on . Therefore, we can say that our ideal approximations by -positive linear operators presented in this paper are more general and applicable than the classical ones.

Finally, for a given , we consider the following subspace of : In this case, we consider the following test functions: Thus, considering the same methods used in this paper, one can immediately get the following ideal approximation result on the subspace , .

Corollary 3.3. *Let be an admissible ideal in , and let be a sequence of -positive linear operators from into itself. Then, for every , ,
**
if and only if, for each ,
*