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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 178316, 12 pages
http://dx.doi.org/10.1155/2012/178316
Research Article

Ideal Convergence of π‘˜-Positive Linear Operators

1Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 9 F. Agaev Street, 1141 Baku, Azerbaijan
2Department of Mathematics, Faculty of Arts and Sciences, TOBB University of Economics and Technology, Sâğütâzü, 06530 Ankara, Turkey

Received 3 March 2010; Accepted 17 June 2010

Academic Editor: MariaΒ Carro

Copyright Β© 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.

Abstract

We study some ideal convergence results of π‘˜-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€œpositivity” 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 β€œπ‘˜-positivity” introduced by GadΕΎiev [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 π‘˜-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 π‘˜-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 π‘˜-positive linear operators which satisfy all conditions of our results but not the classical ones.

We first recall the concept of β€œideal 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)πœ™ (the empty set) βˆˆβ„,(ii)if 𝐴,π΅βˆˆβ„, then 𝐴βˆͺπ΅βˆˆβ„,(iii)if π΄βˆˆβ„ and π΅βŠ†π΄, then π΅βˆˆβ„.

An ideal is called nontrivial if π‘‹βˆ‰β„. Also, a nontrivial ideal in 𝑋 is called admissible if {π‘₯}βˆˆβ„ for each π‘₯βˆˆπ‘‹ (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 β„•, the set of all positive integers. A sequence π‘₯={π‘₯𝑛}π‘›βˆˆβ„• is ideal convergent (or ℐ-convergent) to a number 𝐿 if, for every πœ€>0, {π‘›βˆˆβ„•βˆΆ|π‘₯π‘›βˆ’πΏ|β‰₯πœ€}βˆˆβ„, which is denoted by β„βˆ’lim𝑛π‘₯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 β„•, then ℐ-convergence reduces to the usual. Furthermore, ℐ-convergence coincides with the concept of 𝐴-statistical convergence (see [16]) by taking ℐ={πΎβŠ†β„•βˆΆπ›Ώπ΄(𝐾)=0}, where 𝐴 is a nonnegative regular summability matrix and 𝛿𝐴(𝐾) denotes the 𝐴 density of 𝐾. Besides, if we choose 𝐴=𝐢1, 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 𝐷 be a bounded simply connected domain in β„‚, the set of all complex numbers. By 𝐴(𝐷) we denote the space of all analytic functions on 𝐷. Let πœ™(𝑧) be any analytic function mapping 𝐷 conformally and one to one on the unit disk. Then, for every π‘“βˆˆπ΄(𝐷), we have the Taylor expansion of 𝑓 given by 𝑓(𝑧)=βˆžξ“π‘˜=0π‘“π‘˜(πœ™(𝑧))π‘˜,(1.1) where π‘“π‘˜, π‘˜βˆˆβ„•0=β„•βˆͺ{0}, is the Taylor coefficient of 𝑓 satisfying limsupπ‘˜π‘˜ξ”||π‘“π‘˜||≀1.(1.2) It is known that Taylor’s coefficients are calculated by the following formula (see [18]): π‘“π‘˜=1ξ€Ÿ2πœ‹π‘–πΆπ‘“(𝑧)πœ™ξ…ž(𝑧)(πœ™(𝑧))π‘˜+1𝑑𝑧,(1.3) where 𝐢 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 |πœ™(𝑧)|=π‘Ÿ<1. It is well known that 𝐴(D) is FrΓ©chet’s space with topology of compact convergence in any closed subset of 𝐷. In this paper, we use the norm β€–β‹…β€–π‘Ÿ on the space 𝐴(𝐷) defined by β€–π‘“β€–π‘ŸβˆΆ=max||||πœ™(𝑧)=π‘Ÿ<1||||,𝑓(𝑧)forπ‘“βˆˆπ΄(𝐷),(1.4) and therefore the convergence in 𝐴(𝐷) is the convergence in norm β€–β‹…β€–π‘Ÿ for any 0<π‘Ÿ<1.

Now let 𝐴(𝐷+)∢={π‘“βˆˆπ΄(𝐷)βˆΆπ‘“π‘˜β‰₯0,π‘˜=0,1,2,…}. Following [7] (see also [9]), if a linear operator 𝑇 mapping 𝐴(𝐷) into itself satisfies the condition 𝑇𝐴𝐷+ξ€·π·ξ€Έξ€ΈβŠ†π΄+ξ€Έ,(1.5) then we say that 𝑇 is a β€œπ‘˜-positive linear operator”. As usual, for a function π‘“βˆˆπ΄(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)=βˆžξ“π‘˜=0ξƒ©βˆžξ“π‘š=0π‘‡π‘˜,π‘šπ‘“π‘šξƒͺ(πœ™(𝑧))π‘˜,(1.6) where π‘“π‘š(π‘šβˆˆβ„•0) is the Taylor coefficient of 𝑓 and π‘‡π‘˜,π‘š(π‘˜,π‘šβˆˆβ„•0) is the Taylor coefficient of 𝑇([πœ™(𝑧)]π‘˜). If {𝑇𝑛}π‘›βˆˆβ„• is a sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself, then, as in (1.6), we may write that 𝑇𝑛(𝑓;𝑧)=βˆžξ“π‘˜=0ξƒ©βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šπ‘“π‘šξƒͺ(πœ™(𝑧))π‘˜,foreachπ‘›βˆˆβ„•,(1.7) where, of course, βˆ‘βˆžπ‘š=0𝑇(𝑛)π‘˜,π‘šπ‘“π‘š is the Taylor coefficient of 𝑇𝑛(𝑓) for π‘›βˆˆβ„• and π‘˜βˆˆβ„•0. Observe that 𝑇𝑛 is π‘˜-positive if and only if 𝑇(𝑛)π‘˜,π‘šβ‰₯0 for every π‘›βˆˆβ„• and π‘˜,π‘šβˆˆβ„•0.

Throughout the paper we use the following three test functions: 𝑒𝑖(𝑧)∢=βˆžξ“π‘˜=0π‘˜π‘–(πœ™(𝑧))π‘˜,𝑖=0,1,2.(1.8)

We also consider the following subspace of 𝐴(𝐷): π΄βˆ—ξ€½||𝑓(𝐷)∢=π‘“βˆˆπ΄(𝐷)βˆΆπ‘˜||≀𝑀1+π‘˜2ξ€Έforeveryπ‘˜βˆˆβ„•0andforsomeξ€Ύ.𝑀>0(1.9)

2. Ideal Convergence of π‘˜-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 β„•. Then, one says that a sequence {π‘₯𝑛}π‘›βˆˆβ„• is ideal convergent to a number 𝐿 with degree 0<𝛽≀1 if, for every πœ€>0, ξ‚»||π‘₯π‘›βˆˆβ„•βˆΆπ‘›||βˆ’πΏπ‘›1βˆ’π›½ξ‚Όβ‰₯πœ€βˆˆβ„.(2.1) In this case, we write π‘₯π‘›ξ€·π‘›βˆ’πΏ=β„βˆ’π‘œβˆ’π›½ξ€Έ,asπ‘›βŸΆβˆž.(2.2)

Notice that if we choose, in particular, 𝛽=1 in Definition 2.1, we immediately get the ideal convergence of (π‘₯𝑛) to 𝐿. Observe that, according to Definition 2.1, the degree 𝛽 is controlled by the entries of the sequence {π‘₯𝑛}π‘›βˆˆβ„•.

We first need the following two lemmas.

Lemma 2.2. Let ℐ be an admissible ideal in β„•. Assume further that {𝑓𝑛}π‘›βˆˆβ„• is a sequence of analytic functions on 𝐷 with the Taylor coefficients 𝑓𝑛,π‘˜ for each π‘›βˆˆβ„• and π‘˜βˆˆβ„•0. Then, for 0<𝛽≀1, β€–β€–π‘“π‘›β€–β€–π‘Ÿξ€·π‘›=β„βˆ’π‘œβˆ’π›½ξ€Έ,asπ‘›βŸΆβˆž,(2.3) if and only if there exists a sequence {𝑑𝑛,π‘˜}π‘›βˆˆβ„•,π‘˜βˆˆβ„•0 for which the following conditions hold: ||𝑓𝑛,π‘˜||≀𝑑𝑛,π‘˜,foreveryπ‘›βˆˆβ„•,π‘˜βˆˆβ„•0,(2.4)limsupπ‘˜π‘˜βˆšπ‘‘π‘›,π‘˜β‰€1,foreachfixedπ‘›βˆˆβ„•,(2.5)𝑑𝑛,π‘˜ξ€·π‘›=β„βˆ’π‘œβˆ’π›½ξ€Έ,asπ‘›βŸΆβˆž,foreachfixedπ‘˜βˆˆβ„•0.(2.6)

Proof. Necessity. Assume that (2.3) holds. Then, choosing 𝑑𝑛,π‘˜=|𝑓𝑛,π‘˜|, 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 π‘˜βˆˆβ„•0, πœ€π‘›β€–β€–π‘“βˆΆ=π‘›β€–β€–π‘Ÿ=max||||πœ™(𝑧)=π‘Ÿ<1||𝑓𝑛||𝑛(𝑧)=β„βˆ’π‘œβˆ’π›½ξ€Έ,asπ‘›βŸΆβˆž.(2.7) By (1.3), for any π‘Ÿ<1, we obtain that 𝑑𝑛,π‘˜=||𝑓𝑛,π‘˜||≀1ξ€œ2πœ‹||||πœ™(𝑧)=π‘Ÿ||𝑓𝑛||||πœ™(𝑧)ξ…ž||(𝑧)||||πœ™(𝑧)π‘˜+1||||β‰€πœ€π‘‘π‘§π‘›π‘Ÿπ‘˜,forπ‘›βˆˆβ„•,π‘˜βˆˆβ„•0.(2.8) Hence, it follows from (2.8) that, for a given πœ€>0, ξ‚»π‘‘π‘›βˆˆβ„•βˆΆπ‘›,π‘˜π‘›1βˆ’π›½ξ‚ΌβŠ†ξ‚†πœ€β‰₯πœ€π‘›βˆˆβ„•βˆΆπ‘›π‘›1βˆ’π›½β‰₯π‘Ÿπ‘˜πœ€ξ‚‡.(2.9) By (2.7), we get ξ‚†πœ€π‘›βˆˆβ„•βˆΆπ‘›π‘›1βˆ’π›½β‰₯πœ€ξ…žξ‚‡βˆˆβ„,(2.10) where πœ€ξ…ž=π‘Ÿπ‘˜πœ€. Therefore, from (2.9), we conclude that ξ‚»π‘‘π‘›βˆˆβ„•βˆΆπ‘›,π‘˜π‘›1βˆ’π›½ξ‚Όβ‰₯πœ€βˆˆβ„,(2.11) which gives (2.6).Sufficiency. Assume now that conditions (2.4)–(2.6) hold. It follows from (2.5) that the series βˆ‘βˆžπ‘˜=0𝑑𝑛,π‘˜π‘Ÿπ‘˜ is convergent for any π‘Ÿ<1. Then, for every πœ€>0, there exists a positive natural number 𝑁=𝑁(πœ€) such that βˆžξ“π‘˜=𝑁+1𝑑𝑛,π‘˜π‘Ÿβ‰€πœ€(2.12) holds. Using (2.4) and (2.12) and also considering the fact that β€–β€–π‘“π‘›β€–β€–π‘Ÿβ‰€βˆžξ“π‘˜=0||𝑓𝑛,π‘˜||π‘Ÿπ‘˜β‰€π‘ξ“π‘˜=0𝑑𝑛,π‘˜π‘Ÿπ‘˜+βˆžξ“π‘˜=𝑁+1𝑑𝑛,π‘˜π‘Ÿπ‘˜,(2.13) we get β€–β€–π‘“π‘›β€–β€–π‘Ÿβ‰€πœ€+π‘ξ“π‘˜=0𝑑𝑛,π‘˜π‘Ÿπ‘˜,(2.14) which yields that β€–β€–π‘“π‘›β€–β€–π‘Ÿπ‘›1βˆ’π›½β‰€πœ€π‘›1βˆ’π›½+π‘ξ“π‘˜=0𝑑𝑛,π‘˜π‘Ÿπ‘˜π‘›1βˆ’π›½.(2.15) Since π‘Ÿπ‘˜β‰€1 for every π‘˜=0,1,…,𝑁 and 1/𝑛1βˆ’π›½β‰€1 for every π‘›βˆˆβ„• and 0<𝛽≀1, we obtain that β€–β€–π‘“π‘›β€–β€–π‘Ÿπ‘›1βˆ’π›½β‰€πœ€+π‘ξ“π‘˜=0𝑑𝑛,π‘˜π‘›1βˆ’π›½.(2.16) Now, for a given 𝛾>0, choose an πœ€>0 such that πœ€<𝛾. Then, define the following sets: ξ‚»β€–β€–π‘“πΈβˆΆ=π‘›βˆˆβ„•βˆΆπ‘›β€–β€–π‘Ÿπ‘›1βˆ’π›½ξ‚Ό,𝐸β‰₯π›Ύπ‘˜ξ‚»π‘‘βˆΆ=π‘›βˆˆβ„•βˆΆπ‘›,π‘˜π‘›1βˆ’π›½β‰₯π›Ύβˆ’πœ€ξ‚Όπ‘+1,π‘˜=0,1,…,𝑁.(2.17) From (2.16), it is clear that πΈβŠ†π‘ξšπ‘˜=0πΈπ‘˜.(2.18) By hypothesis (2.6), we know πΈπ‘˜βˆˆβ„ for each π‘˜=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 β„•, and let {𝑇𝑛}π‘›βˆˆβ„• be a sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself. If, for some 𝛽𝑖 with 0<𝛽𝑖≀1(𝑖=0,1,2), ‖‖𝑇𝑛(𝑒𝑖)βˆ’π‘’π‘–β€–β€–π‘Ÿξ€·π‘›=β„βˆ’π‘œβˆ’π›½π‘–ξ€Έ,asπ‘›βŸΆβˆž,(2.19) holds, then there exists a sequence {𝑑𝑛,π‘˜}π‘›βˆˆβ„•,π‘˜βˆˆβ„•0 satisfying (2.5), (2.6) with π›½βˆΆ=min{𝛽0,𝛽1,𝛽2} such that the following inequality βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š(π‘šβˆ’π‘˜)2≀(π‘˜+1)2𝑑𝑛,π‘˜(2.20) holds, where 𝑇(𝑛)π‘˜,π‘š is the same as (1.7).

Proof. We first observe that βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š(π‘šβˆ’π‘˜)2=ξƒ©βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šπ‘š2βˆ’π‘˜2ξƒͺξƒ©βˆ’2π‘˜βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šξƒͺπ‘šβˆ’π‘˜+π‘˜2ξƒ©βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šξƒͺ≀|||||βˆ’1βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šπ‘š2βˆ’π‘˜2||||||||||+2π‘˜βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š|||||π‘šβˆ’π‘˜+π‘˜2|||||βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š|||||.βˆ’1(2.21) By (2.19) and Lemma 2.2, there exist sequences {𝑑𝑖,𝑛,π‘˜}π‘›βˆˆβ„•,π‘˜βˆˆβ„•0 and numbers 𝛽𝑖 with 0<𝛽𝑖≀1(𝑖=0,1,2) satisfying (2.5), (2.6) such that the following conditions |||||βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š|||||βˆ’1≀𝑑0,𝑛,π‘˜,|||||βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š|||||π‘šβˆ’π‘˜β‰€π‘‘1,𝑛,π‘˜,|||||βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šπ‘š2βˆ’π‘˜2|||||≀𝑑2,𝑛,π‘˜(2.22) hold. Hence, we easily get βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š(π‘šβˆ’π‘˜)2≀(π‘˜+1)2𝑑𝑛,π‘˜,(2.23) where 𝑑𝑛,π‘˜βˆΆ=max{𝑑0,𝑛,π‘˜,𝑑1,𝑛,π‘˜,𝑑2,𝑛,π‘˜}. Since π›½βˆΆ=min{𝛽0,𝛽1,𝛽2}, we can see that 𝑑𝑛,π‘˜ξ€·π‘›=π‘œβˆ’π›½ξ€Έ,asπ‘›βŸΆβˆž,(2.24) whence the result.

Then, we are ready to give our main result.

Theorem 2.4. Let ℐ be an admissible ideal in β„•, and let {𝑇𝑛}π‘›βˆˆβ„• be a sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself. If, for some 𝛽𝑖 with 0<𝛽𝑖≀1(𝑖=0,1,2), (2.19) holds, then, for every π‘“βˆˆπ΄βˆ—(𝐷), one has ‖‖𝑇𝑛‖‖(𝑓)βˆ’π‘“π‘Ÿξ€·π‘›=β„βˆ’π‘œ1βˆ’π›½ξ€Έ,asπ‘›βŸΆβˆž,(2.25) where π›½βˆΆ=min{𝛽0,𝛽1,𝛽2}.

Proof. Assume that (2.19) holds for some 𝛽𝑖 with 0<𝛽𝑖≀1(𝑖=0,1,2). Let π‘“βˆˆπ΄βˆ—(𝐷) and π‘§βˆˆπ· be fixed. By (1.1) and (1.7), we may write that 𝑇𝑛(𝑓;𝑧)βˆ’π‘“(𝑧)=βˆžξ“π‘˜=0(πœ™(𝑧))π‘˜βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šξ€·π‘“π‘šβˆ’π‘“π‘˜ξ€Έ+βˆžξ“π‘˜=0(πœ™(𝑧))π‘˜π‘“π‘˜ξƒ©βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘šξƒͺ.βˆ’1(2.26) The last inequality gives that ‖‖𝑇𝑛‖‖(𝑓)βˆ’π‘“π‘Ÿβ‰€βˆžξ“π‘˜=0π‘Ÿπ‘˜βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š||π‘“π‘šβˆ’π‘“π‘˜||+βˆžξ“π‘˜=0π‘Ÿπ‘˜||π‘“π‘˜|||||||βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š|||||.βˆ’1(2.27) By Lemma  2.2(ii) of [11], we know the fact that |π‘“π‘šβˆ’π‘“π‘˜|≀𝑀(3+π‘˜)4(π‘šβˆ’π‘˜)2. Also, since π‘“βˆˆπ΄βˆ—(𝐷), we see that |π‘“π‘˜|≀𝑀(1+π‘˜2). Hence, combining these inequalities, we have ‖‖𝑇𝑛‖‖(𝑓)βˆ’π‘“π‘Ÿβ‰€π‘€βˆžξ“π‘˜=0π‘Ÿπ‘˜(3+π‘˜)4βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š(π‘šβˆ’π‘˜)2+π‘€βˆžξ“π‘˜=0π‘Ÿπ‘˜ξ€·1+π‘˜2ξ€Έ|||||βˆžξ“π‘š=0𝑇(𝑛)π‘˜,π‘š|||||.βˆ’1(2.28) By Lemma 2.3, there exist a sequence {𝑑𝑛,π‘˜}π‘›βˆˆβ„•,π‘˜βˆˆβ„•0 satisfying (2.5), (2.6) with π›½βˆΆ=min{𝛽0,𝛽1,𝛽2} such that the inequality ‖‖𝑇𝑛‖‖(𝑓)βˆ’π‘“π‘Ÿξƒ©β‰€π‘€βˆžξ“π‘˜=0π‘Ÿπ‘˜(3+π‘˜)4(π‘˜+1)2𝑑𝑛,π‘˜+βˆžξ“π‘˜=0π‘Ÿπ‘˜ξ€·1+π‘˜2ξ€Έ(π‘˜+1)2𝑑𝑛,π‘˜ξƒͺ(2.29) holds. Hence, we get, for every π‘›βˆˆβ„•, that ‖‖𝑇𝑛‖‖(𝑓)βˆ’π‘“π‘Ÿβ‰€π‘€βˆžξ“π‘˜=0π‘Ÿπ‘˜(3+π‘˜)6𝑑𝑛,π‘˜.(2.30) Since, for every π‘˜,π‘›βˆˆβ„•, π‘Ÿπ‘˜(3+π‘˜)6𝑑𝑛,π‘˜β‰€π‘Ÿπ‘˜46π‘˜6𝑑𝑛,π‘˜,(2.31) we easily get that ξ€½π‘Ÿπ‘˜(3+π‘˜)6𝑑𝑛,π‘˜ξ€Ύ1/π‘˜β‰€π‘Ÿ46/π‘˜π‘˜6/π‘˜π‘‘1/π‘˜π‘›,π‘˜.(2.32) Now taking limsup as π‘˜β†’βˆž 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<π‘Ÿ<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 |πœ™(𝑧)|=π‘Ÿ<1 with πœ™ξ…ž(𝑧)β‰ 0, 1=1βˆ’πœ™(𝑧)βˆžξ“π‘˜=0(πœ™(𝑧))π‘˜=𝑒0(𝑧),πœ™(𝑧)(1βˆ’πœ™(𝑧))2=πœ™(𝑧)πœ™ξ…žπ‘‘(𝑧)ξ‚΅1𝑑𝑧1βˆ’πœ™(𝑧)=πœ™(𝑧)βˆžξ“π‘˜=0π‘˜(πœ™(𝑧))π‘˜βˆ’1=βˆžξ“π‘˜=0π‘˜(πœ™(𝑧))π‘˜=𝑒1πœ™(𝑧),2(𝑧)(1βˆ’πœ™(𝑧))3=πœ™2(𝑧)2πœ™ξ…žπ‘‘(𝑧)ξ‚΅1𝑑𝑧(1βˆ’πœ™(𝑧))2ξ‚Ά=πœ™2(𝑧)2βˆžξ“π‘˜=0π‘˜(π‘˜βˆ’1)(πœ™(𝑧))π‘˜βˆ’2=12βˆžξ“π‘˜=0π‘˜(π‘˜βˆ’1)(πœ™(𝑧))π‘˜=𝑒2(𝑧)βˆ’π‘’1(𝑧)2.(3.1) Then, defining 𝑓𝑖(𝑧)∢=(πœ™(𝑧))𝑖(1βˆ’πœ™(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 β„•, and let {𝑇𝑛}π‘›βˆˆβ„• be a sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself. If, for some 𝛽𝑖 with 0<𝛽𝑖≀1(𝑖=0,1,2), ‖‖𝑇𝑛(𝑓𝑖)βˆ’π‘“π‘–β€–β€–π‘Ÿξ€·π‘›=β„βˆ’π‘œβˆ’π›½π‘–ξ€Έ,asπ‘›βŸΆβˆž,(3.3) then, for every π‘“βˆˆπ΄βˆ—(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 π‘˜-positive linear operators.

Corollary 3.2. Let ℐ be an admissible ideal in β„•, and let {𝑇𝑛}π‘›βˆˆβ„• be a sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself. Then, for every π‘“βˆˆπ΄βˆ—(𝐷), β„βˆ’lim𝑛‖‖𝑇𝑛‖‖(𝑓)βˆ’π‘“π‘Ÿ=0,(3.4) if and only if, for each 𝑖=0,1,2, β„βˆ’lim𝑛‖‖𝑇𝑛(𝑒𝑖)βˆ’π‘’π‘–β€–β€–π‘Ÿ=0,(3.5) or equivalently, β„βˆ’lim𝑛‖‖𝑇𝑛(𝑓𝑖)βˆ’π‘“π‘–β€–β€–π‘Ÿ=0,(3.6)

Proof. Since each 𝑒𝑖,π‘“π‘–βˆˆπ΄βˆ—(𝐷), 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 ℐ={πΎβŠ†β„•βˆΆπ›Ώπ΄(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={π‘§βˆˆβ„‚βˆΆ|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={πΎβŠ†β„•βˆΆπ›Ώ(𝐾)=0}, where 𝛿(𝐾) denotes the asymptotic density of 𝐾 given by 𝛿(𝐾)∢=lim𝑛#{π‘˜β‰€π‘›βˆΆπ‘˜βˆˆπΎ}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 #{𝐡} we denote the cardinality of the set 𝐡. Hence, let {𝑒𝑛}π‘›βˆˆβ„• be a sequence whose terms are defined by π‘’π‘›ξƒ―π‘›βˆΆ=,𝑛+1if𝑛=π‘š2,π‘šβˆˆβ„•,0,otherwise.(3.8) Then, we easily observe that β„βˆ’lim𝑛𝑒𝑛=stβˆ’lim𝑛𝑒𝑛=0(3.9) holds. Assume now that {𝑇𝑛}π‘›βˆˆβ„• is any sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself, such that, for every π‘“βˆˆπ΄βˆ—(𝐷), the sequence {𝑇𝑛}π‘›βˆˆβ„• is uniformly convergent to 𝑓 on a bounded simply domain 𝐷 with respect to any norm β€–β‹…β€–π‘Ÿ(0<π‘Ÿ<1). Then, consider the following operators: 𝐿𝑛(𝑓;𝑧)∢=1+𝑒𝑛𝑇𝑛(𝑓,𝑧).(3.10) Therefore, observe that {𝐿𝑛}π‘›βˆˆβ„• is a sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself. By (3.10), we can write, for each 𝑖=0,1,2, that ‖‖𝐿𝑛(𝑓𝑖)βˆ’π‘“π‘–β€–β€–π‘Ÿβ‰€β€–β€–π‘‡π‘›(𝑓𝑖)βˆ’π‘“π‘–β€–β€–π‘Ÿ+𝑒𝑛‖‖𝑇𝑛(𝑓𝑖)β€–β€–π‘Ÿβ‰€ξ€·1+𝑒𝑛‖‖𝑇𝑛(𝑓𝑖)βˆ’π‘“π‘–β€–β€–π‘Ÿ+𝑒𝑛‖‖(𝑓𝑖)β€–β€–π‘Ÿβ‰€ξ€·1+𝑒𝑛‖‖𝑇𝑛(𝑓𝑖)βˆ’π‘“π‘–β€–β€–π‘Ÿ+π‘’π‘›π‘Ÿπ‘–(1βˆ’π‘Ÿ)𝑖+1.(3.11) So, the last inequality gives stβˆ’limπ‘›β€–β€–πΏπ‘›ξ€·π‘“π‘–ξ€Έβˆ’π‘“π‘–β€–β€–β‰€ξ‚»stβˆ’lim𝑛1+𝑒𝑛limπ‘›β€–β€–π‘‡π‘›ξ€·π‘’π‘–ξ€Έβˆ’π‘’π‘–β€–β€–ξ‚Ό+stβˆ’limπ‘›π‘’π‘›π‘Ÿπ‘–(1βˆ’π‘Ÿ)𝑖+1(3.12) by our conditions on {𝑇𝑛}π‘›βˆˆβ„•. Then, it follows from (3.9) that, for each 𝑖=0,1,2, stβˆ’limπ‘›β€–β€–πΏπ‘›ξ€·π‘“π‘–ξ€Έβˆ’π‘“π‘–β€–β€–=0.(3.13) Hence, by Corollary 3.2, we obtain, for every π‘“βˆˆπ΄βˆ—(𝐷), that β„βˆ’lim𝑛‖‖𝐿𝑛‖‖=(𝑓)βˆ’π‘“stβˆ’lim𝑛‖‖𝐿𝑛‖‖(𝑓)βˆ’π‘“=0.(3.14) However, by the definition (3.8), we see that, for every π‘“βˆˆπ΄βˆ—(𝐷), 𝑒𝑛‖‖𝑇𝑛‖‖(𝑓)π‘Ÿ=𝑛‖‖𝑇𝑛+1𝑛‖‖(𝑓)π‘Ÿ,if𝑛=π‘š2,π‘šβˆˆβ„•,0,otherwise.(3.15) Now since lim𝑛‖𝑇𝑛(𝑓)β€–π‘Ÿ=β€–π‘“β€–π‘Ÿ, we immediately obtain that the subsequence {𝑒𝑛‖𝑇𝑛(𝑓)β€–π‘Ÿ}𝑛=π‘š2 converges to β€–π‘“β€–π‘Ÿ while the subsequence {𝑒𝑛‖𝑇𝑛(𝑓)β€–π‘Ÿ}π‘›β‰ π‘š2 converges to zero. Hence, the sequence {𝑒𝑛‖𝑇𝑛(𝑓)β€–π‘Ÿ}π‘›βˆˆβ„• is nonconvergent. Therefore, we see that the sequence {𝐿𝑛(𝑓)}π‘›βˆˆβ„• cannot be uniformly convergent to 𝑓 on 𝐷. Therefore, we can say that our ideal approximations by π‘˜-positive linear operators presented in this paper are more general and applicable than the classical ones.

Finally, for a given π‘šβˆˆβ„•, we consider the following subspace of 𝐴(𝐷): π΄βˆ—π‘šξ€½||𝑓(𝐷)∢=π‘“βˆˆπ΄(𝐷)βˆΆπ‘˜||≀𝑀1+π‘˜2π‘šξ€Έforeveryπ‘˜βˆˆβ„•0andforsomeξ€Ύ.𝑀>0(3.16) In this case, we consider the following test functions: 𝑔𝑖(𝑧)=βˆžξ“π‘˜=0π‘˜π‘šπ‘–(πœ™(𝑧))π‘˜,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 π΄βˆ—π‘š(𝐷), π‘šβˆˆβ„•.

Corollary 3.3. Let ℐ be an admissible ideal in β„•, and let {𝑇𝑛}π‘›βˆˆβ„• be a sequence of π‘˜-positive linear operators from 𝐴(𝐷) into itself. Then, for every π‘“βˆˆπ΄βˆ—π‘š(𝐷), π‘šβˆˆβ„•, β„βˆ’lim𝑛‖‖𝑇𝑛‖‖(𝑓)βˆ’π‘“π‘Ÿ=0,(3.18) if and only if, for each 𝑖=0,1,2, β„βˆ’lim𝑛‖‖𝑇𝑛(𝑔𝑖)βˆ’π‘”π‘–β€–β€–π‘Ÿ=0.(3.19)

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