Ideal Convergence of 𝑘-Positive Linear Operators

Abstract

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 [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 Korovkin-type 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 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 ℐ of subsets of 𝑋 is said to be an ideal in 𝑋 provided that(i)𝜙 (the empty set) ∈ℐ,(ii)if 𝐴,𝐵∈ℐ, then 𝐴∪𝐵∈ℐ,(iii)if 𝐴∈ℐ and 𝐵⊆𝐴, then 𝐵∈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∶|𝑥_𝑛−𝐿|≥𝜀}∈ℐ, which is denoted by ℐ−lim_𝑛𝑥_𝑛=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 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=𝐶₁, 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 𝑓(𝑧)=^∞_𝑘=0𝑓_𝑘(𝜙(𝑧))^𝑘,(1.1) where 𝑓_𝑘, 𝑘∈ℕ₀=ℕ∪{0}, is the Taylor coefficient of 𝑓 satisfying limsup_𝑘^𝑘||𝑓_𝑘||≤1.(1.2) It is known that Taylor’s coefficients are calculated by the following formula (see [18]): 𝑓_𝑘=12𝜋𝑖_𝐶𝑓(𝑧)𝜙^(𝑧)(𝜙(𝑧))^𝑘+1𝑑𝑧,(1.3) 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 ‖𝑓‖_𝑟∶=max_{||||𝜙(𝑧)=𝑟<1}||𝑓(𝑧)||,for𝑓∈𝐴(𝐷),(1.4) and therefore the convergence in 𝐴(𝐷) is the convergence in norm ‖⋅‖_𝑟 for any 0<𝑟<1.

Now let 𝐴(𝐷⁺)∶={𝑓∈𝐴(𝐷)∶𝑓_𝑘≥0,𝑘=0,1,2,…}. Following [7] (see also [9]), if a linear operator 𝑇 mapping 𝐴(𝐷) into itself satisfies the condition 𝑇𝐴𝐷⁺⊆𝐴𝐷⁺,(1.5) 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 𝑇(𝑓;𝑧)=^∞_𝑘=0^∞_𝑚=0𝑇_𝑘,𝑚𝑓_𝑚(𝜙(𝑧))^𝑘,(1.6) where 𝑓_𝑚(𝑚∈ℕ₀) is the Taylor coefficient of 𝑓 and 𝑇_𝑘,𝑚(𝑘,𝑚∈ℕ₀) is the Taylor coefficient of 𝑇([𝜙(𝑧)]^𝑘). If {𝑇_𝑛}_𝑛∈ℕ is a sequence of 𝑘-positive linear operators from 𝐴(𝐷) into itself, then, as in (1.6), we may write that 𝑇_𝑛(𝑓;𝑧)=^∞_𝑘=0^∞_𝑚=0𝑇^{(𝑛)𝑘,𝑚}𝑓_𝑚(𝜙(𝑧))^𝑘,foreach𝑛∈ℕ,(1.7) where, of course, ∑^∞𝑚=0𝑇^{(𝑛)𝑘,𝑚}𝑓_𝑚 is the Taylor coefficient of 𝑇_𝑛(𝑓) for 𝑛∈ℕ and 𝑘∈ℕ₀. Observe that 𝑇_𝑛 is 𝑘-positive if and only if 𝑇^{(𝑛)𝑘,𝑚}≥0 for every 𝑛∈ℕ and 𝑘,𝑚∈ℕ₀.

Throughout the paper we use the following three test functions: 𝑒_𝑖(𝑧)∶=^∞_𝑘=0𝑘^𝑖(𝜙(𝑧))^𝑘,𝑖=0,1,2.(1.8)

We also consider the following subspace of 𝐴(𝐷): 𝐴^∗(𝐷)∶=𝑓∈𝐴(𝐷)∶||𝑓_𝑘||≤𝑀1+𝑘²forevery𝑘∈ℕ₀andforsome𝑀>0.(1.9)

2. 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, 𝑛∈ℕ∶||𝑥^𝑛−𝐿||𝑛^1−𝛽≥𝜀∈ℐ.(2.1) In this case, we write 𝑥_𝑛−𝐿=ℐ−𝑜𝑛^−𝛽,as𝑛⟶∞.(2.2)

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∈ℕ₀. Then, for 0<𝛽≤1, ‖‖𝑓_𝑛‖‖_𝑟=ℐ−𝑜𝑛^−𝛽,as𝑛⟶∞,(2.3) if and only if there exists a sequence {𝑡_𝑛,𝑘}_{𝑛∈ℕ,𝑘∈ℕ0} for which the following conditions hold: ||𝑓_𝑛,𝑘||≤𝑡_𝑛,𝑘,forevery𝑛∈ℕ,𝑘∈ℕ₀,(2.4)limsup√_𝑘^𝑘𝑡_𝑛,𝑘≤1,foreachfixed𝑛∈ℕ,(2.5)𝑡_𝑛,𝑘=ℐ−𝑜𝑛^−𝛽,as𝑛⟶∞,foreachfixed𝑘∈ℕ₀.(2.6)

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∈ℕ₀, 𝜀_𝑛∶=‖‖𝑓_𝑛‖‖_𝑟=max_{||||𝜙(𝑧)=𝑟<1}||𝑓_𝑛(𝑧)||=ℐ−𝑜𝑛^−𝛽,as𝑛⟶∞.(2.7) By (1.3), for any 𝑟<1, we obtain that 𝑡_𝑛,𝑘=||𝑓_𝑛,𝑘||≤12𝜋_{||||𝜙(𝑧)=𝑟}||𝑓_𝑛(𝑧)||||𝜙^(𝑧)||||𝜙(𝑧)||^𝑘+1||𝑑𝑧||≤𝜀^𝑛𝑟^𝑘,for𝑛∈ℕ,𝑘∈ℕ₀.(2.8) Hence, it follows from (2.8) that, for a given 𝜀>0, 𝑛∈ℕ∶𝑛𝑡^{𝑛,𝑘1−𝛽}≥𝜀⊆𝑛∈ℕ∶𝜀^𝑛𝑛^1−𝛽≥𝑟^𝑘𝜀.(2.9) By (2.7), we get 𝑛∈ℕ∶𝑛𝜀^𝑛_1−𝛽≥𝜀^∈ℐ,(2.10) where 𝜀^=𝑟^𝑘𝜀. Therefore, from (2.9), we conclude that 𝑛∈ℕ∶𝑛𝑡^{𝑛,𝑘1−𝛽}≥𝜀∈ℐ,(2.11) which gives (2.6).Sufficiency. Assume now that conditions (2.4)–(2.6) hold. It follows from (2.5) that the series ∑^∞𝑘=0𝑡_𝑛,𝑘𝑟^𝑘 is convergent for any 𝑟<1. Then, for every 𝜀>0, there exists a positive natural number 𝑁=𝑁(𝜀) such that ^∞_𝑘_=𝑁+1𝑡_𝑛,𝑘𝑟≤𝜀(2.12) holds. Using (2.4) and (2.12) and also considering the fact that ‖‖𝑓_𝑛‖‖_𝑟≤^∞_𝑘=0||𝑓_𝑛,𝑘||𝑟^𝑘≤^𝑁_𝑘=0𝑡_𝑛,𝑘𝑟^𝑘+^∞_𝑘=_𝑁+1𝑡_𝑛,𝑘𝑟^𝑘,(2.13) we get ‖‖𝑓_𝑛‖‖_𝑟≤𝜀+^𝑁_𝑘=0𝑡_𝑛,𝑘𝑟^𝑘,(2.14) which yields that ‖‖𝑓_𝑛‖‖_𝑟𝑛^1−𝛽≤𝜀𝑛^1−𝛽+^𝑁_𝑘=0𝑡_𝑛,𝑘𝑟^𝑘𝑛^1−𝛽.(2.15) Since 𝑟^𝑘≤1 for every 𝑘=0,1,…,𝑁 and 1/𝑛^1−𝛽≤1 for every 𝑛∈ℕ and 0<𝛽≤1, we obtain that ‖‖𝑓_𝑛‖‖_𝑟𝑛^1−𝛽≤𝜀+^𝑁_𝑘=0𝑡_𝑛,𝑘𝑛^1−𝛽.(2.16) Now, for a given 𝛾>0, choose an 𝜀>0 such that 𝜀<𝛾. Then, define the following sets: 𝐸∶=𝑛∈ℕ∶‖‖𝑓^𝑛‖‖_𝑟𝑛^1−𝛽𝐸≥𝛾_𝑘,∶=𝑛∈ℕ∶𝑡^𝑛,𝑘𝑛^1−𝛽≥𝛾−𝜀𝑁+1,𝑘=0,1,…,𝑁.(2.17) From (2.16), it is clear that 𝐸⊆^𝑁_𝑘=0𝐸_𝑘.(2.18) 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 𝛽_𝑖 with 0<𝛽_𝑖≤1(𝑖=0,1,2), ‖‖𝑇_𝑛(𝑒_𝑖)−𝑒_𝑖‖‖_𝑟=ℐ−𝑜𝑛^−𝛽𝑖,as𝑛⟶∞,(2.19) holds, then there exists a sequence {𝑡_𝑛,𝑘}_{𝑛∈ℕ,𝑘∈ℕ0} satisfying (2.5), (2.6) with 𝛽∶=min{𝛽₀,𝛽₁,𝛽₂} such that the following inequality ^∞_𝑚=0𝑇^{(𝑛)𝑘,𝑚}(𝑚−𝑘)²≤(𝑘+1)²𝑡_𝑛,𝑘(2.20) holds, where 𝑇^{(𝑛)𝑘,𝑚} is the same as (1.7).

Proof. We first observe that ^∞_𝑚=0𝑇^{(𝑛)𝑘,𝑚}(𝑚−𝑘)²=^∞_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}𝑚²−𝑘²−2𝑘^∞_𝑚=0𝑇^{(𝑛)𝑘,𝑚}𝑚−𝑘+𝑘²^∞_𝑚=0≤|||||𝑇^{(𝑛)𝑘,𝑚}−1_∞_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}𝑚²−𝑘²|||||+2𝑘|||||_∞𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}𝑚−𝑘|||||+𝑘²|||||_∞𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}−1|||||.(2.21) By (2.19) and Lemma 2.2, there exist sequences {𝑡_{𝑖,𝑛,𝑘}}_{𝑛∈ℕ,𝑘∈ℕ0} and numbers 𝛽_𝑖 with 0<𝛽_𝑖≤1(𝑖=0,1,2) satisfying (2.5), (2.6) such that the following conditions |||||_∞𝑚₌₀|||||_∞𝑇^{(𝑛)𝑘,𝑚}−1|||||≤𝑡_0,𝑛,𝑘,_𝑚₌₀|||||_∞𝑇^{(𝑛)𝑘,𝑚}𝑚−𝑘|||||≤𝑡_1,𝑛,𝑘,_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}𝑚²−𝑘²|||||≤𝑡_2,𝑛,𝑘(2.22) hold. Hence, we easily get ^∞_𝑚=0𝑇^{(𝑛)𝑘,𝑚}(𝑚−𝑘)²≤(𝑘+1)²𝑡_𝑛,𝑘,(2.23) where 𝑡_𝑛,𝑘∶=max{𝑡_0,𝑛,𝑘,𝑡_1,𝑛,𝑘,𝑡_2,𝑛,𝑘}. Since 𝛽∶=min{𝛽₀,𝛽₁,𝛽₂}, we can see that 𝑡_𝑛,𝑘=𝑜𝑛^−𝛽,as𝑛⟶∞,(2.24) whence the result.

Then, we are ready to give our main result.

Theorem 2.4. 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), (2.19) holds, then, for every 𝑓∈𝐴^∗(𝐷), one has ‖‖𝑇_𝑛(𝑓)−𝑓‖‖_𝑟=ℐ−𝑜𝑛^1−𝛽,as𝑛⟶∞,(2.25) where 𝛽∶=min{𝛽₀,𝛽₁,𝛽₂}.

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 𝑇_𝑛(𝑓;𝑧)−𝑓(𝑧)=^∞_𝑘=0(𝜙(𝑧))^𝑘∞_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}𝑓_𝑚−𝑓_𝑘+^∞_𝑘=0(𝜙(𝑧))^𝑘𝑓_𝑘^∞_𝑚=0𝑇^{(𝑛)𝑘,𝑚}−1.(2.26) The last inequality gives that ‖‖𝑇_𝑛(𝑓)−𝑓‖‖_𝑟≤^∞_𝑘=0𝑟^𝑘∞_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}||𝑓_𝑚−𝑓_𝑘||+^∞_𝑘=0𝑟^𝑘||𝑓_𝑘|||||||^∞_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}−1|||||.(2.27) By Lemma  2.2(ii) of [11], we know the fact that |𝑓_𝑚−𝑓_𝑘|≤𝑀(3+𝑘)⁴(𝑚−𝑘)². Also, since 𝑓∈𝐴^∗(𝐷), we see that |𝑓_𝑘|≤𝑀(1+𝑘²). Hence, combining these inequalities, we have ‖‖𝑇_𝑛(𝑓)−𝑓‖‖_𝑟≤𝑀^∞_𝑘=0𝑟^𝑘(3+𝑘)^4∞_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}(𝑚−𝑘)²+𝑀^∞_𝑘=0𝑟^𝑘1+𝑘²|||||^∞_𝑚₌₀𝑇^{(𝑛)𝑘,𝑚}−1|||||.(2.28) By Lemma 2.3, there exist a sequence {𝑡_𝑛,𝑘}_{𝑛∈ℕ,𝑘∈ℕ0} satisfying (2.5), (2.6) with 𝛽∶=min{𝛽₀,𝛽₁,𝛽₂} such that the inequality ‖‖𝑇_𝑛(𝑓)−𝑓‖‖_𝑟≤𝑀^∞_𝑘=0𝑟^𝑘(3+𝑘)⁴(𝑘+1)²𝑡_𝑛,𝑘+^∞_𝑘=0𝑟^𝑘1+𝑘²(𝑘+1)²𝑡_𝑛,𝑘(2.29) holds. Hence, we get, for every 𝑛∈ℕ, that ‖‖𝑇_𝑛(𝑓)−𝑓‖‖_𝑟≤𝑀^∞_𝑘=0𝑟^𝑘(3+𝑘)⁶𝑡_𝑛,𝑘.(2.30) Since, for every 𝑘,𝑛∈ℕ, 𝑟^𝑘(3+𝑘)⁶𝑡_𝑛,𝑘≤𝑟^𝑘4⁶𝑘⁶𝑡_𝑛,𝑘,(2.31) we easily get that 𝑟^𝑘(3+𝑘)⁶𝑡_𝑛,𝑘^1/𝑘≤𝑟4^6/𝑘𝑘^6/𝑘𝑡^{1/𝑘𝑛,𝑘}.(2.32) 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=0𝜙(𝑧)(𝜙(𝑧))^𝑘=𝑒₀(𝑧),(1−𝜙(𝑧))²=𝜙(𝑧)𝜙^(𝑧)𝑑𝑑𝑧1−1𝜙(𝑧)=𝜙(𝑧)^∞_𝑘=0𝑘(𝜙(𝑧))^𝑘−1=^∞_𝑘=0𝜙𝑘(𝜙(𝑧))^𝑘=𝑒₁(𝑧),₂(𝑧)(1−𝜙(𝑧))³=𝜙²(𝑧)2𝜙^(𝑧)𝑑𝑑𝑧(1−𝜙1(𝑧))²=𝜙²(𝑧)2^∞_𝑘=0𝑘(𝑘−1)(𝜙(𝑧))^𝑘−2=12^∞_𝑘=0𝑘(𝑘−1)(𝜙(𝑧))^𝑘=𝑒²(𝑧)−𝑒₁(𝑧)2.(3.1) Then, defining 𝑓_𝑖(𝑧)∶=(𝜙(𝑧))^𝑖(1−𝜙(𝑧))^𝑖+1,𝑖=0,1,2,(3.2) 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), ‖‖𝑇_𝑛(𝑓_𝑖)−𝑓_𝑖‖‖_𝑟=ℐ−𝑜𝑛^−𝛽𝑖,as𝑛⟶∞,(3.3) 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.

Corollary 3.2. 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,(3.5) or equivalently, ℐ−lim_𝑛‖‖𝑇_𝑛(𝑓_𝑖)−𝑓_𝑖‖‖_𝑟=0,(3.6)

Proof. 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.

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(providedthelimitexists),(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 𝑢_𝑛∶=𝑛+𝑛1,0,if𝑛=𝑚²,𝑚∈ℕ,otherwise.(3.8) Then, we easily observe that ℐ−lim_𝑛𝑢_𝑛=st−lim_𝑛𝑢_𝑛=0(3.9) 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: 𝐿_𝑛(𝑓;𝑧)∶=1+𝑢_𝑛𝑇_𝑛(𝑓,𝑧).(3.10) 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 ‖‖𝐿_𝑛(𝑓_𝑖)−𝑓_𝑖‖‖_𝑟≤‖‖𝑇_𝑛(𝑓_𝑖)−𝑓_𝑖‖‖_𝑟+𝑢_𝑛‖‖𝑇_𝑛(𝑓_𝑖)‖‖_𝑟≤1+𝑢_𝑛‖‖𝑇_𝑛(𝑓_𝑖)−𝑓_𝑖‖‖_𝑟+𝑢_𝑛‖‖(𝑓_𝑖)‖‖_𝑟≤1+𝑢_𝑛‖‖𝑇_𝑛(𝑓_𝑖)−𝑓_𝑖‖‖_𝑟+𝑢^𝑛𝑟^𝑖(1−𝑟)^𝑖+1.(3.11) So, the last inequality gives st−lim_𝑛‖‖𝐿_𝑛𝑓_𝑖−𝑓_𝑖‖‖≤st−lim_𝑛1+𝑢_𝑛lim_𝑛‖‖𝑇_𝑛𝑒_𝑖−𝑒_𝑖‖‖+st−lim_𝑛𝑢_𝑛𝑟^𝑖(1−𝑟)^𝑖+1(3.12) by our conditions on {𝑇_𝑛}_𝑛∈ℕ. Then, it follows from (3.9) that, for each 𝑖=0,1,2, st−lim_𝑛‖‖𝐿_𝑛𝑓_𝑖−𝑓_𝑖‖‖=0.(3.13) Hence, by Corollary 3.2, we obtain, for every 𝑓∈𝐴^∗(𝐷), that ℐ−lim_𝑛‖‖𝐿_𝑛(𝑓)−𝑓‖‖=st−lim_𝑛‖‖𝐿_𝑛(𝑓)−𝑓‖‖=0.(3.14) However, by the definition (3.8), we see that, for every 𝑓∈𝐴^∗(𝐷), 𝑢_𝑛‖‖𝑇_𝑛(𝑓)‖‖_𝑟=𝑛+𝑛1‖‖𝑇^𝑛0,(𝑓)‖‖_𝑟,if𝑛=𝑚²,𝑚∈ℕ,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=𝑚2 converges to ‖𝑓‖_𝑟 while the subsequence {𝑢_𝑛‖𝑇_𝑛(𝑓)‖_𝑟}_𝑛≠𝑚2 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forevery𝑘∈ℕ₀andforsome𝑀>0.(3.16) In this case, we consider the following test functions: 𝑔_𝑖(𝑧)=^∞_𝑘=0𝑘^𝑚𝑖(𝜙(𝑧))^𝑘,𝑖=0,1,2.(3.17) Thus, considering the same methods used in this paper, one can immediately get the following ideal approximation result on the subspace 𝐴^∗𝑚(𝐷), 𝑚∈ℕ.

Corollary 3.3. 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. Then, for every 𝑓∈𝐴^∗𝑚(𝐷), 𝑚∈ℕ, ℐ−lim_𝑛‖‖𝑇_𝑛(𝑓)−𝑓‖‖_𝑟=0,(3.18) if and only if, for each 𝑖=0,1,2, ℐ−lim_𝑛‖‖𝑇_𝑛(𝑔_𝑖)−𝑔_𝑖‖‖_𝑟=0.(3.19)