We study some ideal convergence results of š‘˜-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 ā€œpositivityā€ of real-valued linear operators. In [2], this theory was improved by Gadjiev and Orhan via the concept of statistical convergence (see, also, [3ā€“6]). However, in order to obtain the Korovkin-type results for complex-valued linear operators, the concept of ā€œš‘˜-positivityā€ introduced by Gadžiev [7] is used instead of the classical positivity. Such results may be found in the papers [7ā€“10]. In the recent works [11, 12], with the help of some convergence methods, such as š“-statistical convergence and ideal convergence, various approximation theorems have been obtained for š‘˜-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 š‘˜-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 š‘˜-positive linear operators which satisfy all conditions of our results but not the classical ones.

We first recall the concept of ā€œideal convergence.ā€

Let š‘‹ be a nonempty set. A class ā„ of subsets of š‘‹ is said to be an ideal in š‘‹ provided that(i)šœ™ (the empty set) āˆˆā„,(ii)if š“,šµāˆˆā„, then š“āˆŖšµāˆˆā„,(iii)if š“āˆˆā„ and šµāŠ†š“, then šµāˆˆā„.

An ideal is called nontrivial if š‘‹āˆ‰ā„. Also, a nontrivial ideal in š‘‹ is called admissible if {š‘„}āˆˆā„ for each š‘„āˆˆš‘‹ (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 ā„ be a nontrivial ideal in ā„•, the set of all positive integers. A sequence š‘„={š‘„š‘›}š‘›āˆˆā„• is ideal convergent (or ā„-convergent) to a number šæ if, for every šœ€>0, {š‘›āˆˆā„•āˆ¶|š‘„š‘›āˆ’šæ|ā‰„šœ€}āˆˆā„, which is denoted by ā„āˆ’limš‘›š‘„š‘›=šæ. Notice that the method of ideal convergence includes many convergence methods such as š“-statistical convergence, statistical convergence, lacunary statistical convergence, usual convergence, and so forth. For example, if ā„ is the class of all finite subsets of ā„•, then ā„-convergence reduces to the usual. Furthermore, ā„-convergence coincides with the concept of š“-statistical convergence (see [16]) by taking ā„={š¾āŠ†ā„•āˆ¶š›æš“(š¾)=0}, where š“ is a nonnegative regular summability matrix and š›æš“(š¾) denotes the š“ density of š¾. Besides, if we choose š“=š¶1, the CesĆ”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 š· be a bounded simply connected domain in ā„‚, the set of all complex numbers. By š“(š·) we denote the space of all analytic functions on š·. Let šœ™(š‘§) be any analytic function mapping š· conformally and one to one on the unit disk. Then, for every š‘“āˆˆš“(š·), we have the Taylor expansion of š‘“ given by š‘“(š‘§)=āˆžī“š‘˜=0š‘“š‘˜(šœ™(š‘§))š‘˜,(1.1) where š‘“š‘˜, š‘˜āˆˆā„•0=ā„•āˆŖ{0}, is the Taylor coefficient of š‘“ satisfying limsupš‘˜š‘˜ī”||š‘“š‘˜||ā‰¤1.(1.2) It is known that Taylorā€™s coefficients are calculated by the following formula (see [18]): š‘“š‘˜=1ī€Ÿ2šœ‹š‘–š¶š‘“(š‘§)šœ™ī…ž(š‘§)(šœ™(š‘§))š‘˜+1š‘‘š‘§,(1.3) where š¶ is any closed contour lying in the interior of š·. It is not hard to see that the series (1.1) under the condition (1.2) is uniformly convergent if |šœ™(š‘§)|=š‘Ÿ<1. It is well known that š“(D) is FrĆ©chetā€™s space with topology of compact convergence in any closed subset of š·. In this paper, we use the norm ā€–ā‹…ā€–š‘Ÿ on the space š“(š·) defined by ā€–š‘“ā€–š‘Ÿāˆ¶=max||||šœ™(š‘§)=š‘Ÿ<1||||,š‘“(š‘§)forš‘“āˆˆš“(š·),(1.4) and therefore the convergence in š“(š·) is the convergence in norm ā€–ā‹…ā€–š‘Ÿ for any 0<š‘Ÿ<1.

Now let š“(š·+)āˆ¶={š‘“āˆˆš“(š·)āˆ¶š‘“š‘˜ā‰„0,š‘˜=0,1,2,ā€¦}. Following [7] (see also [9]), if a linear operator š‘‡ mapping š“(š·) into itself satisfies the condition š‘‡ī€·š“ī€·š·+ī€·š·ī€øī€øāŠ†š“+ī€ø,(1.5) then we say that š‘‡ is a ā€œš‘˜-positive linear operatorā€. As usual, for a function š‘“āˆˆš“(š·), the value of š‘‡(š‘“) at a point š‘§ is denoted by š‘‡(š‘“;š‘§) and also the Taylor expansion of š‘‡(š‘“) is š‘‡(š‘“;š‘§)=āˆžī“š‘˜=0īƒ©āˆžī“š‘š=0š‘‡š‘˜,š‘šš‘“š‘šīƒŖ(šœ™(š‘§))š‘˜,(1.6) where š‘“š‘š(š‘šāˆˆā„•0) is the Taylor coefficient of š‘“ and š‘‡š‘˜,š‘š(š‘˜,š‘šāˆˆā„•0) is the Taylor coefficient of š‘‡([šœ™(š‘§)]š‘˜). If {š‘‡š‘›}š‘›āˆˆā„• is a sequence of š‘˜-positive linear operators from š“(š·) into itself, then, as in (1.6), we may write that š‘‡š‘›(š‘“;š‘§)=āˆžī“š‘˜=0īƒ©āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šš‘“š‘šīƒŖ(šœ™(š‘§))š‘˜,foreachš‘›āˆˆā„•,(1.7) where, of course, āˆ‘āˆžš‘š=0š‘‡(š‘›)š‘˜,š‘šš‘“š‘š is the Taylor coefficient of š‘‡š‘›(š‘“) for š‘›āˆˆā„• and š‘˜āˆˆā„•0. Observe that š‘‡š‘› is š‘˜-positive if and only if š‘‡(š‘›)š‘˜,š‘šā‰„0 for every š‘›āˆˆā„• and š‘˜,š‘šāˆˆā„•0.

Throughout the paper we use the following three test functions: š‘’š‘–(š‘§)āˆ¶=āˆžī“š‘˜=0š‘˜š‘–(šœ™(š‘§))š‘˜,š‘–=0,1,2.(1.8)

We also consider the following subspace of š“(š·): š“āˆ—ī€½||š‘“(š·)āˆ¶=š‘“āˆˆš“(š·)āˆ¶š‘˜||ī€·ā‰¤š‘€1+š‘˜2ī€øforeveryš‘˜āˆˆā„•0andforsomeī€¾.š‘€>0(1.9)

2. Ideal Convergence of š‘˜-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 ā„ be an admissible ideal in ā„•. Then, one says that a sequence {š‘„š‘›}š‘›āˆˆā„• is ideal convergent to a number šæ with degree 0<š›½ā‰¤1 if, for every šœ€>0, ī‚»||š‘„š‘›āˆˆā„•āˆ¶š‘›||āˆ’šæš‘›1āˆ’š›½ī‚¼ā‰„šœ€āˆˆā„.(2.1) In this case, we write š‘„š‘›ī€·š‘›āˆ’šæ=ā„āˆ’š‘œāˆ’š›½ī€ø,asš‘›āŸ¶āˆž.(2.2)

Notice that if we choose, in particular, š›½=1 in Definition 2.1, we immediately get the ideal convergence of (š‘„š‘›) to šæ. Observe that, according to Definition 2.1, the degree š›½ is controlled by the entries of the sequence {š‘„š‘›}š‘›āˆˆā„•.

We first need the following two lemmas.

Lemma 2.2. Let ā„ be an admissible ideal in ā„•. Assume further that {š‘“š‘›}š‘›āˆˆā„• is a sequence of analytic functions on š· with the Taylor coefficients š‘“š‘›,š‘˜ for each š‘›āˆˆā„• and š‘˜āˆˆā„•0. Then, for 0<š›½ā‰¤1, ā€–ā€–š‘“š‘›ā€–ā€–š‘Ÿī€·š‘›=ā„āˆ’š‘œāˆ’š›½ī€ø,asš‘›āŸ¶āˆž,(2.3) if and only if there exists a sequence {š‘”š‘›,š‘˜}š‘›āˆˆā„•,š‘˜āˆˆā„•0 for which the following conditions hold: ||š‘“š‘›,š‘˜||ā‰¤š‘”š‘›,š‘˜,foreveryš‘›āˆˆā„•,š‘˜āˆˆā„•0,(2.4)limsupš‘˜š‘˜āˆšš‘”š‘›,š‘˜ā‰¤1,foreachļ¬xedš‘›āˆˆā„•,(2.5)š‘”š‘›,š‘˜ī€·š‘›=ā„āˆ’š‘œāˆ’š›½ī€ø,asš‘›āŸ¶āˆž,foreachļ¬xedš‘˜āˆˆā„•0.(2.6)

Proof. Necessity. Assume that (2.3) holds. Then, choosing š‘”š‘›,š‘˜=|š‘“š‘›,š‘˜|, 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 š‘˜āˆˆā„•0, šœ€š‘›ā€–ā€–š‘“āˆ¶=š‘›ā€–ā€–š‘Ÿ=max||||šœ™(š‘§)=š‘Ÿ<1||š‘“š‘›||ī€·š‘›(š‘§)=ā„āˆ’š‘œāˆ’š›½ī€ø,asš‘›āŸ¶āˆž.(2.7) By (1.3), for any š‘Ÿ<1, we obtain that š‘”š‘›,š‘˜=||š‘“š‘›,š‘˜||ā‰¤1ī€œ2šœ‹||||šœ™(š‘§)=š‘Ÿ||š‘“š‘›||||šœ™(š‘§)ī…ž||(š‘§)||||šœ™(š‘§)š‘˜+1||||ā‰¤šœ€š‘‘š‘§š‘›š‘Ÿš‘˜,forš‘›āˆˆā„•,š‘˜āˆˆā„•0.(2.8) Hence, it follows from (2.8) that, for a given šœ€>0, ī‚»š‘”š‘›āˆˆā„•āˆ¶š‘›,š‘˜š‘›1āˆ’š›½ī‚¼āŠ†ī‚†šœ€ā‰„šœ€š‘›āˆˆā„•āˆ¶š‘›š‘›1āˆ’š›½ā‰„š‘Ÿš‘˜šœ€ī‚‡.(2.9) By (2.7), we get ī‚†šœ€š‘›āˆˆā„•āˆ¶š‘›š‘›1āˆ’š›½ā‰„šœ€ī…žī‚‡āˆˆā„,(2.10) where šœ€ī…ž=š‘Ÿš‘˜šœ€. Therefore, from (2.9), we conclude that ī‚»š‘”š‘›āˆˆā„•āˆ¶š‘›,š‘˜š‘›1āˆ’š›½ī‚¼ā‰„šœ€āˆˆā„,(2.11) which gives (2.6).Sufficiency. Assume now that conditions (2.4)ā€“(2.6) hold. It follows from (2.5) that the series āˆ‘āˆžš‘˜=0š‘”š‘›,š‘˜š‘Ÿš‘˜ is convergent for any š‘Ÿ<1. Then, for every šœ€>0, there exists a positive natural number š‘=š‘(šœ€) such that āˆžī“š‘˜=š‘+1š‘”š‘›,š‘˜š‘Ÿā‰¤šœ€(2.12) holds. Using (2.4) and (2.12) and also considering the fact that ā€–ā€–š‘“š‘›ā€–ā€–š‘Ÿā‰¤āˆžī“š‘˜=0||š‘“š‘›,š‘˜||š‘Ÿš‘˜ā‰¤š‘ī“š‘˜=0š‘”š‘›,š‘˜š‘Ÿš‘˜+āˆžī“š‘˜=š‘+1š‘”š‘›,š‘˜š‘Ÿš‘˜,(2.13) we get ā€–ā€–š‘“š‘›ā€–ā€–š‘Ÿā‰¤šœ€+š‘ī“š‘˜=0š‘”š‘›,š‘˜š‘Ÿš‘˜,(2.14) which yields that ā€–ā€–š‘“š‘›ā€–ā€–š‘Ÿš‘›1āˆ’š›½ā‰¤šœ€š‘›1āˆ’š›½+š‘ī“š‘˜=0š‘”š‘›,š‘˜š‘Ÿš‘˜š‘›1āˆ’š›½.(2.15) Since š‘Ÿš‘˜ā‰¤1 for every š‘˜=0,1,ā€¦,š‘ and 1/š‘›1āˆ’š›½ā‰¤1 for every š‘›āˆˆā„• and 0<š›½ā‰¤1, we obtain that ā€–ā€–š‘“š‘›ā€–ā€–š‘Ÿš‘›1āˆ’š›½ā‰¤šœ€+š‘ī“š‘˜=0š‘”š‘›,š‘˜š‘›1āˆ’š›½.(2.16) Now, for a given š›¾>0, choose an šœ€>0 such that šœ€<š›¾. Then, define the following sets: ī‚»ā€–ā€–š‘“šøāˆ¶=š‘›āˆˆā„•āˆ¶š‘›ā€–ā€–š‘Ÿš‘›1āˆ’š›½ī‚¼,šøā‰„š›¾š‘˜ī‚»š‘”āˆ¶=š‘›āˆˆā„•āˆ¶š‘›,š‘˜š‘›1āˆ’š›½ā‰„š›¾āˆ’šœ€ī‚¼š‘+1,š‘˜=0,1,ā€¦,š‘.(2.17) From (2.16), it is clear that šøāŠ†š‘īšš‘˜=0šøš‘˜.(2.18) By hypothesis (2.6), we know šøš‘˜āˆˆā„ for each š‘˜=0,1,ā€¦,š‘. Hence, by the definition of ideal, we immediately obtain that the set šø belongs to ā„ for every š›¾>0, which implies (2.3). So, the proof is completed.

Lemma 2.3. Let ā„ be an admissible ideal in ā„•, and let {š‘‡š‘›}š‘›āˆˆā„• be a sequence of š‘˜-positive linear operators from š“(š·) into itself. If, for some š›½š‘– with 0<š›½š‘–ā‰¤1(š‘–=0,1,2), ā€–ā€–š‘‡š‘›(š‘’š‘–)āˆ’š‘’š‘–ā€–ā€–š‘Ÿī€·š‘›=ā„āˆ’š‘œāˆ’š›½š‘–ī€ø,asš‘›āŸ¶āˆž,(2.19) holds, then there exists a sequence {š‘”š‘›,š‘˜}š‘›āˆˆā„•,š‘˜āˆˆā„•0 satisfying (2.5), (2.6) with š›½āˆ¶=min{š›½0,š›½1,š›½2} such that the following inequality āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š(š‘šāˆ’š‘˜)2ā‰¤(š‘˜+1)2š‘”š‘›,š‘˜(2.20) holds, where š‘‡(š‘›)š‘˜,š‘š is the same as (1.7).

Proof. We first observe that āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š(š‘šāˆ’š‘˜)2=īƒ©āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šš‘š2āˆ’š‘˜2īƒŖīƒ©āˆ’2š‘˜āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šīƒŖš‘šāˆ’š‘˜+š‘˜2īƒ©āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šīƒŖā‰¤|||||āˆ’1āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šš‘š2āˆ’š‘˜2||||||||||+2š‘˜āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š|||||š‘šāˆ’š‘˜+š‘˜2|||||āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š|||||.āˆ’1(2.21) By (2.19) and Lemma 2.2, there exist sequences {š‘”š‘–,š‘›,š‘˜}š‘›āˆˆā„•,š‘˜āˆˆā„•0 and numbers š›½š‘– with 0<š›½š‘–ā‰¤1(š‘–=0,1,2) satisfying (2.5), (2.6) such that the following conditions |||||āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š|||||āˆ’1ā‰¤š‘”0,š‘›,š‘˜,|||||āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š|||||š‘šāˆ’š‘˜ā‰¤š‘”1,š‘›,š‘˜,|||||āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šš‘š2āˆ’š‘˜2|||||ā‰¤š‘”2,š‘›,š‘˜(2.22) hold. Hence, we easily get āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š(š‘šāˆ’š‘˜)2ā‰¤(š‘˜+1)2š‘”š‘›,š‘˜,(2.23) where š‘”š‘›,š‘˜āˆ¶=max{š‘”0,š‘›,š‘˜,š‘”1,š‘›,š‘˜,š‘”2,š‘›,š‘˜}. Since š›½āˆ¶=min{š›½0,š›½1,š›½2}, we can see that š‘”š‘›,š‘˜ī€·š‘›=š‘œāˆ’š›½ī€ø,asš‘›āŸ¶āˆž,(2.24) whence the result.

Then, we are ready to give our main result.

Theorem 2.4. Let ā„ be an admissible ideal in ā„•, and let {š‘‡š‘›}š‘›āˆˆā„• be a sequence of š‘˜-positive linear operators from š“(š·) into itself. If, for some š›½š‘– with 0<š›½š‘–ā‰¤1(š‘–=0,1,2), (2.19) holds, then, for every š‘“āˆˆš“āˆ—(š·), one has ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)āˆ’š‘“š‘Ÿī€·š‘›=ā„āˆ’š‘œ1āˆ’š›½ī€ø,asš‘›āŸ¶āˆž,(2.25) where š›½āˆ¶=min{š›½0,š›½1,š›½2}.

Proof. Assume that (2.19) holds for some š›½š‘– with 0<š›½š‘–ā‰¤1(š‘–=0,1,2). Let š‘“āˆˆš“āˆ—(š·) and š‘§āˆˆš· be fixed. By (1.1) and (1.7), we may write that š‘‡š‘›(š‘“;š‘§)āˆ’š‘“(š‘§)=āˆžī“š‘˜=0(šœ™(š‘§))š‘˜āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šī€·š‘“š‘šāˆ’š‘“š‘˜ī€ø+āˆžī“š‘˜=0(šœ™(š‘§))š‘˜š‘“š‘˜īƒ©āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘šīƒŖ.āˆ’1(2.26) The last inequality gives that ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)āˆ’š‘“š‘Ÿā‰¤āˆžī“š‘˜=0š‘Ÿš‘˜āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š||š‘“š‘šāˆ’š‘“š‘˜||+āˆžī“š‘˜=0š‘Ÿš‘˜||š‘“š‘˜|||||||āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š|||||.āˆ’1(2.27) By Lemmaā€‰ā€‰2.2(ii) of [11], we know the fact that |š‘“š‘šāˆ’š‘“š‘˜|ā‰¤š‘€(3+š‘˜)4(š‘šāˆ’š‘˜)2. Also, since š‘“āˆˆš“āˆ—(š·), we see that |š‘“š‘˜|ā‰¤š‘€(1+š‘˜2). Hence, combining these inequalities, we have ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)āˆ’š‘“š‘Ÿā‰¤š‘€āˆžī“š‘˜=0š‘Ÿš‘˜(3+š‘˜)4āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š(š‘šāˆ’š‘˜)2+š‘€āˆžī“š‘˜=0š‘Ÿš‘˜ī€·1+š‘˜2ī€ø|||||āˆžī“š‘š=0š‘‡(š‘›)š‘˜,š‘š|||||.āˆ’1(2.28) By Lemma 2.3, there exist a sequence {š‘”š‘›,š‘˜}š‘›āˆˆā„•,š‘˜āˆˆā„•0 satisfying (2.5), (2.6) with š›½āˆ¶=min{š›½0,š›½1,š›½2} such that the inequality ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)āˆ’š‘“š‘Ÿīƒ©ā‰¤š‘€āˆžī“š‘˜=0š‘Ÿš‘˜(3+š‘˜)4(š‘˜+1)2š‘”š‘›,š‘˜+āˆžī“š‘˜=0š‘Ÿš‘˜ī€·1+š‘˜2ī€ø(š‘˜+1)2š‘”š‘›,š‘˜īƒŖ(2.29) holds. Hence, we get, for every š‘›āˆˆā„•, that ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)āˆ’š‘“š‘Ÿā‰¤š‘€āˆžī“š‘˜=0š‘Ÿš‘˜(3+š‘˜)6š‘”š‘›,š‘˜.(2.30) Since, for every š‘˜,š‘›āˆˆā„•, š‘Ÿš‘˜(3+š‘˜)6š‘”š‘›,š‘˜ā‰¤š‘Ÿš‘˜46š‘˜6š‘”š‘›,š‘˜,(2.31) we easily get that ī€½š‘Ÿš‘˜(3+š‘˜)6š‘”š‘›,š‘˜ī€¾1/š‘˜ā‰¤š‘Ÿ46/š‘˜š‘˜6/š‘˜š‘”1/š‘˜š‘›,š‘˜.(2.32) Now taking limsup as š‘˜ā†’āˆž 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<š‘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 |šœ™(š‘§)|=š‘Ÿ<1 with šœ™ī…ž(š‘§)ā‰ 0, 1=1āˆ’šœ™(š‘§)āˆžī“š‘˜=0(šœ™(š‘§))š‘˜=š‘’0(š‘§),šœ™(š‘§)(1āˆ’šœ™(š‘§))2=šœ™(š‘§)šœ™ī…žš‘‘(š‘§)ī‚µ1š‘‘š‘§ī‚¶1āˆ’šœ™(š‘§)=šœ™(š‘§)āˆžī“š‘˜=0š‘˜(šœ™(š‘§))š‘˜āˆ’1=āˆžī“š‘˜=0š‘˜(šœ™(š‘§))š‘˜=š‘’1šœ™(š‘§),2(š‘§)(1āˆ’šœ™(š‘§))3=šœ™2(š‘§)2šœ™ī…žš‘‘(š‘§)ī‚µ1š‘‘š‘§(1āˆ’šœ™(š‘§))2ī‚¶=šœ™2(š‘§)2āˆžī“š‘˜=0š‘˜(š‘˜āˆ’1)(šœ™(š‘§))š‘˜āˆ’2=12āˆžī“š‘˜=0š‘˜(š‘˜āˆ’1)(šœ™(š‘§))š‘˜=š‘’2(š‘§)āˆ’š‘’1(š‘§)2.(3.1) Then, defining š‘“š‘–(š‘§)āˆ¶=(šœ™(š‘§))š‘–(1āˆ’šœ™(š‘§))š‘–+1,š‘–=0,1,2,(3.2) the next result is equivalent to Theorem 2.4.

Theorem 3.1. Let ā„ be an admissible ideal in ā„•, and let {š‘‡š‘›}š‘›āˆˆā„• be a sequence of š‘˜-positive linear operators from š“(š·) into itself. If, for some š›½š‘– with 0<š›½š‘–ā‰¤1(š‘–=0,1,2), ā€–ā€–š‘‡š‘›(š‘“š‘–)āˆ’š‘“š‘–ā€–ā€–š‘Ÿī€·š‘›=ā„āˆ’š‘œāˆ’š›½š‘–ī€ø,asš‘›āŸ¶āˆž,(3.3) then, for every š‘“āˆˆš“āˆ—(š·), (2.25) holds for the same š›½ as in Theorem 2.4.

If we take š›½=1 in Theorems 2.4 and 3.1, then we immediately get the following characterization for ideal approximation by š‘˜-positive linear operators.

Corollary 3.2. Let ā„ be an admissible ideal in ā„•, and let {š‘‡š‘›}š‘›āˆˆā„• be a sequence of š‘˜-positive linear operators from š“(š·) into itself. Then, for every š‘“āˆˆš“āˆ—(š·), ā„āˆ’limš‘›ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)āˆ’š‘“š‘Ÿ=0,(3.4) if and only if, for each š‘–=0,1,2, ā„āˆ’limš‘›ā€–ā€–š‘‡š‘›(š‘’š‘–)āˆ’š‘’š‘–ā€–ā€–š‘Ÿ=0,(3.5) or equivalently, ā„āˆ’limš‘›ā€–ā€–š‘‡š‘›(š‘“š‘–)āˆ’š‘“š‘–ā€–ā€–š‘Ÿ=0,(3.6)

Proof. Since each š‘’š‘–,š‘“š‘–āˆˆš“āˆ—(š·), the implications (3.4) ā‡’ (3.5) and (3.4) ā‡’ (3.6) are obvious. The sufficiency immediately follows from Definition 2.1 and Theorems 2.4 and 3.1.

Finally, if we choose ā„={š¾āŠ†ā„•āˆ¶š›æš“(š¾)=0}, where š“ is a nonnegative regular summability matrix and š›æš“(š¾) denotes the š“ density of š¾, and also if we take š·={š‘§āˆˆā„‚āˆ¶|š‘§|<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 ā„={š¾āŠ†ā„•āˆ¶š›æ(š¾)=0}, where š›æ(š¾) denotes the asymptotic density of š¾ given by š›æ(š¾)āˆ¶=limš‘›#{š‘˜ā‰¤š‘›āˆ¶š‘˜āˆˆš¾}š‘›(providedthelimitexists),(3.7) then ā„-convergence reduces to the concept of statistical convergence which was first introduced by Fast [17]. In the last equality, by #{šµ} we denote the cardinality of the set šµ. Hence, let {š‘¢š‘›}š‘›āˆˆā„• be a sequence whose terms are defined by š‘¢š‘›īƒÆš‘›āˆ¶=,š‘›+1ifš‘›=š‘š2,š‘šāˆˆā„•,0,otherwise.(3.8) Then, we easily observe that ā„āˆ’limš‘›š‘¢š‘›=stāˆ’limš‘›š‘¢š‘›=0(3.9) holds. Assume now that {š‘‡š‘›}š‘›āˆˆā„• is any sequence of š‘˜-positive linear operators from š“(š·) into itself, such that, for every š‘“āˆˆš“āˆ—(š·), the sequence {š‘‡š‘›}š‘›āˆˆā„• is uniformly convergent to š‘“ on a bounded simply domain š· with respect to any norm ā€–ā‹…ā€–š‘Ÿ(0<š‘Ÿ<1). Then, consider the following operators: šæš‘›ī€·(š‘“;š‘§)āˆ¶=1+š‘¢š‘›ī€øš‘‡š‘›(š‘“,š‘§).(3.10) Therefore, observe that {šæš‘›}š‘›āˆˆā„• is a sequence of š‘˜-positive linear operators from š“(š·) into itself. By (3.10), we can write, for each š‘–=0,1,2, that ā€–ā€–šæš‘›(š‘“š‘–)āˆ’š‘“š‘–ā€–ā€–š‘Ÿā‰¤ā€–ā€–š‘‡š‘›(š‘“š‘–)āˆ’š‘“š‘–ā€–ā€–š‘Ÿ+š‘¢š‘›ā€–ā€–š‘‡š‘›(š‘“š‘–)ā€–ā€–š‘Ÿā‰¤ī€·1+š‘¢š‘›ī€øā€–ā€–š‘‡š‘›(š‘“š‘–)āˆ’š‘“š‘–ā€–ā€–š‘Ÿ+š‘¢š‘›ā€–ā€–(š‘“š‘–)ā€–ā€–š‘Ÿā‰¤ī€·1+š‘¢š‘›ī€øā€–ā€–š‘‡š‘›(š‘“š‘–)āˆ’š‘“š‘–ā€–ā€–š‘Ÿ+š‘¢š‘›š‘Ÿš‘–(1āˆ’š‘Ÿ)š‘–+1.(3.11) So, the last inequality gives stāˆ’limš‘›ā€–ā€–šæš‘›ī€·š‘“š‘–ī€øāˆ’š‘“š‘–ā€–ā€–ā‰¤ī‚»stāˆ’limš‘›ī€·1+š‘¢š‘›ī€øī‚¼ī‚»limš‘›ā€–ā€–š‘‡š‘›ī€·š‘’š‘–ī€øāˆ’š‘’š‘–ā€–ā€–ī‚¼+stāˆ’limš‘›š‘¢š‘›š‘Ÿš‘–(1āˆ’š‘Ÿ)š‘–+1(3.12) by our conditions on {š‘‡š‘›}š‘›āˆˆā„•. Then, it follows from (3.9) that, for each š‘–=0,1,2, stāˆ’limš‘›ā€–ā€–šæš‘›ī€·š‘“š‘–ī€øāˆ’š‘“š‘–ā€–ā€–=0.(3.13) Hence, by Corollary 3.2, we obtain, for every š‘“āˆˆš“āˆ—(š·), that ā„āˆ’limš‘›ā€–ā€–šæš‘›ā€–ā€–=(š‘“)āˆ’š‘“stāˆ’limš‘›ā€–ā€–šæš‘›ā€–ā€–(š‘“)āˆ’š‘“=0.(3.14) However, by the definition (3.8), we see that, for every š‘“āˆˆš“āˆ—(š·), š‘¢š‘›ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)š‘Ÿ=īƒÆš‘›ā€–ā€–š‘‡š‘›+1š‘›ā€–ā€–(š‘“)š‘Ÿ,ifš‘›=š‘š2,š‘šāˆˆā„•,0,otherwise.(3.15) Now since limš‘›ā€–š‘‡š‘›(š‘“)ā€–š‘Ÿ=ā€–š‘“ā€–š‘Ÿ, we immediately obtain that the subsequence {š‘¢š‘›ā€–š‘‡š‘›(š‘“)ā€–š‘Ÿ}š‘›=š‘š2 converges to ā€–š‘“ā€–š‘Ÿ while the subsequence {š‘¢š‘›ā€–š‘‡š‘›(š‘“)ā€–š‘Ÿ}š‘›ā‰ š‘š2 converges to zero. Hence, the sequence {š‘¢š‘›ā€–š‘‡š‘›(š‘“)ā€–š‘Ÿ}š‘›āˆˆā„• is nonconvergent. Therefore, we see that the sequence {šæš‘›(š‘“)}š‘›āˆˆā„• cannot be uniformly convergent to š‘“ on š·. Therefore, we can say that our ideal approximations by š‘˜-positive linear operators presented in this paper are more general and applicable than the classical ones.

Finally, for a given š‘šāˆˆā„•, we consider the following subspace of š“(š·): š“āˆ—š‘šī€½||š‘“(š·)āˆ¶=š‘“āˆˆš“(š·)āˆ¶š‘˜||ī€·ā‰¤š‘€1+š‘˜2š‘šī€øforeveryš‘˜āˆˆā„•0andforsomeī€¾.š‘€>0(3.16) In this case, we consider the following test functions: š‘”š‘–(š‘§)=āˆžī“š‘˜=0š‘˜š‘šš‘–(šœ™(š‘§))š‘˜,š‘–=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 š“āˆ—š‘š(š·), š‘šāˆˆā„•.

Corollary 3.3. Let ā„ be an admissible ideal in ā„•, and let {š‘‡š‘›}š‘›āˆˆā„• be a sequence of š‘˜-positive linear operators from š“(š·) into itself. Then, for every š‘“āˆˆš“āˆ—š‘š(š·), š‘šāˆˆā„•, ā„āˆ’limš‘›ā€–ā€–š‘‡š‘›ā€–ā€–(š‘“)āˆ’š‘“š‘Ÿ=0,(3.18) if and only if, for each š‘–=0,1,2, ā„āˆ’limš‘›ā€–ā€–š‘‡š‘›(š‘”š‘–)āˆ’š‘”š‘–ā€–ā€–š‘Ÿ=0.(3.19)