Research Article | Open Access
Ideal Convergence of -Positive Linear Operators
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.
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, [3–6]). However, in order to obtain the Korovkin-type results for complex-valued linear operators, the concept of “-positivity” introduced by Gadžiev  is used instead of the classical positivity. Such results may be found in the papers [7–10]. 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 “ideal convergence.”
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  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 ) by taking , where is a nonnegative regular summability matrix and denotes the density of . Besides, if we choose , 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 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 Taylor’s coefficients are calculated by the following formula (see ): 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 Fréchet’s 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  (see also ), if a linear operator mapping into itself satisfies the condition then we say that is a “-positive linear operator”. 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 ).
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)–(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 Lemma 2.2(ii) of , 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.
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,
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 .
It is known from  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 . 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 ,
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