Ideal Convergence of k-Positive Linear Operators

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.


Introduction
The classical Korovkin theory see 1 is mainly based on the "positivity" of real-valued linear operators.In 2 , this theory was improved by Gadjiev and Orhan via the concept of statistical convergence see, also, 3-6 .However, in order to obtain the Korovkintype results for complex-valued linear operators, the concept of "k-positivity" introduced by Gadžiev 7 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 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 kpositive linear operators which satisfy all conditions of our results but not the classical ones.
We first recall the concept of "ideal convergence."

Journal of Function Spaces and Applications
Let X be a nonempty set.A class I of subsets of X is said to be an ideal in X provided that i φ the empty set ∈ I, An ideal is called nontrivial if X / ∈ I. Also, a nontrivial ideal in X is called admissible if {x} ∈ I for each x ∈ X 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 I be a nontrivial ideal in N, the set of all positive integers.A sequence x {x n } n∈N is ideal convergent or I-convergent to a number L if, for every ε > 0, {n ∈ N : 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 I is the class of all finite subsets of N, then I-convergence reduces to the usual.Furthermore, I-convergence coincides with the concept of A-statistical convergence see 16 by taking I {K ⊆ N : δ A K 0}, where A is a nonnegative regular summability matrix and δ A K denotes the A density of K. Besides, if we choose A C 1 , the Cesáro 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 D be a bounded simply connected domain in C, 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 f ∈ A D , we have the Taylor expansion of f given by It is known that Taylor's coefficients are calculated by the following formula see 18 : then we say that T is a "k-positive linear operator".As usual, for a function f ∈ A 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 where as in 1.6 , we may write that Throughout the paper we use the following three test functions: We also consider the following subspace of A D : for every k ∈ N 0 and for some M > 0 . 1.9

Ideal Convergence of k-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 I be an admissible ideal in N.Then, one says that a sequence {x n } n∈N is ideal convergent to a number L with degree 0 < β ≤ 1 if, for every ε > 0, In this case, we write Notice that if we choose, in particular, β 1 in Definition 2.1, we immediately get the ideal convergence of x n to L. Observe that, according to Definition 2.1, the degree β is controlled by the entries of the sequence {x n } n∈N .
We first need the following two lemmas.
Lemma 2.2.Let I be an admissible ideal in N. Assume further that {f n } n∈N is a sequence of analytic functions on D with the Taylor coefficients f n,k for each n ∈ N and k ∈ N 0 .Then, for 0 < β ≤ 1, if and only if there exists a sequence {t n,k } n∈N,k∈N 0 for which the following conditions hold: Proof.
Necessity.Assume that 2.3 holds.Then, choosing t n,k |f n,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 k ∈ N 0 , By 1.3 , for any r < 1, we obtain that Hence, it follows from 2.8 that, for a given ε > 0, By 2.7 , we get where ε r k ε.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 ∞ k 0 t n,k r k is convergent for any r < 1.Then, for every ε > 0, there exists a positive natural number N N ε such that 12 holds.Using 2.4 and 2.12 and also considering the fact that we get

2.17
From 2.16 , it is clear that By hypothesis 2.6 , we know E k ∈ I for each k 0, 1, . . ., N. Hence, by the definition of ideal, we immediately obtain that the set E belongs to I for every γ > 0, which implies 2.3 .So, the proof is completed.

Lemma 2.3. Let I be an admissible ideal in N, and let {T n } n∈N be a sequence of k-positive linear operators from A D into itself. If, for some
holds, then there exists a sequence {t n,k } n∈N,k∈N 0 satisfying 2.5 , 2.6 with β : min{β 0 , β 1 , β 2 } such that the following inequality holds, where T n k,m is the same as 1.7 .
Proof.We first observe that

2.21
By 2.19 and Lemma 2.2, there exist sequences {t i,n,k } n∈N,k∈N 0 and numbers β i with 0 < β i ≤ 1 i 0, 1, 2 satisfying 2.5 , 2.6 such that the following conditions where Proof.Assume that 2.19 holds for some β i with 0 < β i ≤ 1 i 0, 1, 2 .Let f ∈ A * D and z ∈ D be fixed.By 1.1 and 1.7 , we may write that

2.26
The last inequality gives that

2.27
By Lemma 2.2 ii of 11 , we know the fact that Hence, combining these inequalities, we have

2.28
By Lemma 2.3, there exist a sequence {t n,k } n∈N,k∈N 0 satisfying 2.5 , 2.6 with β : min{β 0 , β 1 , β 2 } such that the inequality holds.Hence, we get, for every n ∈ N, that Since, for every k, n ∈ N, we easily get that

2.32
Now taking lim sup 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.

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

3.1
Then, defining the next result is equivalent to Theorem 2.4.
Theorem 3.1.Let I be an admissible ideal in N, and let {T n } n∈N be a sequence of k-positive linear operators from A D into itself.If, for some β i with 0 < β i ≤ 1 i 0, 1, 2 , then, for every f ∈ A * 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.Finally, if we choose I {K ⊆ N : δ A K 0}, where A is a nonnegative regular summability matrix and δ A K denotes the A density of K, and also if we take D {z ∈ C : |z| < 1}, 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 I {K ⊆ N : δ K 0}, where δ K denotes the asymptotic density of K given by δ K : lim n #{k ≤ n : k ∈ K} n provided the limit exists , 3.7 then I-convergence reduces to the concept of statistical convergence which was first introduced by Fast 17 .In the last equality, by #{B} we denote the cardinality of the set B. Hence, let {u n } n∈N be a sequence whose terms are defined by Then, we easily observe that holds.Assume now that {T n } n∈N is any sequence of k-positive linear operators from A D into itself, such that, for every f ∈ A * D , the sequence {T n } n∈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: Therefore, observe that {L n } n∈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

3.11
Journal of Function Spaces and Applications 11 So, the last inequality gives 12 by our conditions on {T n } n∈N .Then, it follows from 3.9 that, for each i 0, 1, 2, Hence, by Corollary 3.2, we obtain, for every f ∈ A * D , that However, by the definition 3.8 , we see that, for every f ∈ A * D , T n f r , if n m 2 , m ∈ N, 0, otherwise.

3.15
Now since lim n T n f r f r , we immediately obtain that the subsequence {u n T n f r } n m 2 converges to f r while the subsequence {u n T n f r } n / m 2 converges to zero.Hence, the sequence {u n T n f r } n∈N is nonconvergent.Therefore, we see that the sequence {L n f } n∈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 ∈ N, we consider the following subspace of A D : A * m D : f ∈ A D : f k ≤ M 1 k 2m for every k ∈ N 0 and for some M > 0 .

3.16
In this case, we consider the following test functions: 3.17

Corollary 3 . 2 .Proof.
Let I be an admissible ideal in N, and let {T n } n∈N be a sequence of k-positive linear operators from A D into itself.Then, for every f ∈ A * D , I − lim n T n f − f r 0, 3.4 if and only if, for each i 0, 1, 2, I − lim n T n e i − e i r 0, Since each e i , f i ∈ A * 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.

Thus, considering theCorollary 3 . 3 .
same methods used in this paper, one can immediately get the following ideal approximation result on the subspace A * m D , m ∈ N. Let I be an admissible ideal in N, and let {T n } n∈N be a sequence of k-positive linear operators from A D into itself.Then, for every f ∈ A * m D , m ∈ N, I − lim n T n f − f r 0, 3.18 3where 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