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Journal of Function Spaces
Volume 2015 (2015), Article ID 160401, 11 pages
http://dx.doi.org/10.1155/2015/160401
Research Article

Korovkin-Type Theorems for Modular --Statistical Convergence

1Department of Mathematics and Computer Science, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2Department of Mathematics, Faculty of Arts and Sciences, Sinop University, 57000 Sinop, Turkey

Received 9 October 2014; Accepted 27 December 2014

Academic Editor: Dragan Djordjevic

Copyright © 2015 Carlo Bardaro 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 deal with a new type of statistical convergence for double sequences, called --statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.

1. Introduction

The classical Bohman-Korovkin theorem (see [14]) establishes the uniform convergence in the space of all continuous real functions defined on the interval , for a sequence of positive linear operators acting on , assuming the convergence only on the test functions , , and . For functions belonging to other functional spaces, the problem to extend the Korovkin theorem is much more difficult and, in general, it holds for functions belonging to a suitable subspace. We quote here [59] for the Korovkin theorem in -spaces, [10, 11] for an extension to Orlicz spaces, [1219] for general modular spaces, and [20, 21] for a comprehensive survey about these topics. In recent years, general approaches to the Korovkin theorem were studied, especially in the direction of statistical convergence, which was first introduced independently by Fast and Steinhaus (see [22, 23]). Another classical paper on the emergence of statistical convergence is [24], and then this kind of convergence was developed by many authors (see, e.g., [17, 2531]). In [32], we have introduced a more general notion of statistical convergence for double sequences of positive linear operators , named “triangular -statistical convergence,” and we have obtained a version of the Korovkin theorem in the space of all continuous real functions defined in a compact subset .

This paper continues and develops the theory introduced in [18, 19, 32, 33]. Here, we consider the more general case of the --statistical convergence, which essentially considers suitable submatrices of the infinite matrix (see Section 2), defined by means of a suitable function . The triangular -statistical convergence is defined by the particular choice . The general definition using an arbitrary function was already announced in [32]. However, here, we consider the general case when the operators , , are acting on an abstract modular function space generated by a modular functional and is a locally compact topological space of finite Borel measure, endowed with a uniform structure. We obtain, in a suitable subspace of , a Korovkin theorem and also we study certain quantitative versions, under suitable assumptions.

Our general approach enables one to unify various extensions of the Korovkin theorem, by choosing suitably the function , the matrix , and the modular functional . In particular, corresponding results in , Orlicz, and Musielak-Orlicz spaces are obtained. In Section 4, we also discuss an extension to nonpositive operators, using an approach introduced in [25] and used also in [12]. Finally, some examples are given, in the setting of Mellin-type convolution operators, which are extensively studied in recent years by many authors (see, e.g., [15, 3441]).

2. Preliminaries

Let be a two-dimensional infinite matrix. Given a double sequence of real numbers, set provided that the series is convergent. We say that is regular if it maps every convergent sequence into a convergent sequence with the same limit. We now recall the Silverman-Toeplitz conditions, which are a characterization of regular two-dimensional matrix transformations (see also [25, 42, 43]):(i);(ii) for each ;(iii).

Let be a nonnegative regular matrix, a fixed function, , and : , , and let denote the cardinality of , for each . The --density of is defined by provided that the limit on the right-hand side exists in .

It is not difficult to check that the --density satisfies the following properties:(i)if , then ;(ii), provided that has a - density.

Definition 1. Let be a nonnegative regular summability matrix. The double sequence is said to be --statistically convergent to if for any one gets where , and we write -.

Given a double sequence in , put and define Observe that and satisfy the usual properties of the limit superior and the limit inferior (see also [12, 44]).

If we take , we obtain the notion of triangular -statistical convergence (see also [32]). Observe that if is the Cesàro matrix, defined by setting then the --statistical convergence can be considered as a generalized concept of the classical statistical convergence.

The -density is defined by The double sequence is said to be -statistically convergent to if for each the set has -density zero, and we write .

We denote by the set of all --statistically convergent sequences.

Remark 2. (a) It is easy to see that if , then --statistical convergence implies --statistical convergence.
Note that, in general, the converse implication is not true. Indeed, let be the Cesàro matrix, a fixed real number, and any double sequence with for each , and put , ,  . We get , and, hence, So the double sequence does not --statistically converge to . On the other hand, for any double sequence , we have . So we get and thus --statistically converges to .
(b) Observe that if is the Cesàro matrix, then is a double sequence, Pringsheim convergent to a real number , namely, such that for every there is with whenever (see also [45, 46]), then is -convergent to for every function . Indeed, if we choose arbitrarily such a and an , then, for each , we have . Thus, we get and hence ; namely,   -converges to . Note that the converse, in general, is not true (see also [32, Example 4]).

3. The Korovkin Theorem in Modular Spaces

Let be a locally compact Hausdorff topological space, endowed with a uniform structure which generates the topology of (see also [47]). Let be the -algebra of all Borel subsets of and a positive finite regular measure defined on . We denote by the space of all real-valued -measurable functions on with identification up to sets of measure zero, by the space of all real-valued continuous and bounded functions on , and by the subspace of of all functions with compact support on .

We now recall the notion of modular space (see also [15, 48]).

A functional is called a modular on if it satisfies the following conditions:(i)   -almost everywhere on ;(ii) for every ;(iii) for every , and for each , with ; a modular is said to be convex if it satisfies conditions (i), (ii),(iii′) for all and for every , with .

Let be a constant. We say that a modular is -quasi-semiconvex if for all , , and (see also [14]).

The modular space generated by is defined by and the space of the finite elements of is We will recall the following properties of modulars (see also [15, 4850]).(a)A modular is monotone if for all with .(b)A modular is finite if (the characteristic function associated with ) belongs to whenever with .(c)A modular is strongly finite if belongs to for all with .(d)A modular is said to be absolutely continuous if there is a positive constant with the property: for all with ,(i)for each , there exists a set with and ,(ii)for every , there is a with for every with .

Example 3 (see [15, 48]). Let be the set of all continuous nondecreasing functions with , for any , and in the usual sense, and let be the set of all elements of which are convex functions.
For every (resp., ), the functional , defined by is a modular (resp., convex) on and is the Orlicz space generated by .

We now define the modular and strong convergences in the context of the --statistical convergence (for the classical case and filter convergence, see [12, 15, 36, 37, 48], resp.).

A double sequence of functions in is -modularly convergent to if there is a with A double sequence in is -strongly convergent to if for every .

For , we obtain the corresponding notions for the “triangular” modular and strong convergences.

Given a subset and , we say that (i.e., is in the modular closure of ) if there is a sequence in such that is modularly convergent to in the usual sense.

We recall the following.

Proposition 4 (see also [51, Theorem 1]). Let be a monotone, strongly finite, and absolutely continuous modular on . Then, with respect to the modular convergence in the ordinary sense.

Throughout this paper, we will consider some kinds of rates of approximation associated with the Korovkin theorem in the context of --statistical convergence, and, for technical reasons, we will sometimes suppose that is a metric space, satisfying the following property: for every and , with , there are points , , such that , , and for each .

Observe that the Euclidean multidimensional spaces endowed with the usual metric fulfil , as well as the space endowed with the sup-norm, where is any abstract nonempty set.

For and , let be the usual modulus of continuity of . Observe that is an increasing function of , for each , , and for every , where and for every , (see also [21]).

We now prove some Korovkin-type theorems with respect to an abstract finite set of test functions in the context of the --statistical convergence.

Let be a double sequence of linear operators , , with . Here, the set is the domain of the operators .

We say that the double sequence , together with the modular , satisfies the property if there exist a subset with and a positive real constant with for any    and , and for every and .

Some examples of operators satisfying property - can be found in [14].

Set for every ; let , , and , , be functions in . Put and assume that (P1) for all ,(P2)for every neighborhood there is a positive real number with whenever , (see also [12]).

From now on, we will assume that , . This happens, for example, when is an open bounded subset of or, more generally, a space of finite measure .

We now prove the following.

Theorem 5. Let be a strongly finite, monotone, and -quasi-semiconvex modular. Assume that and , , satisfy (P1) and (P2). Let , , be a double sequence of positive linear operators with property -. If is -modularly convergent to in for each , then is -modularly convergent to in for every .
If is -strongly convergent to , , in , then is -strongly convergent to in for every .

Proof. Let . Since is endowed with the uniformity , is uniformly continuous and bounded on . Fix arbitrarily . By uniform continuity of , there is an element with whenever , .
For each , let be as in (19), and, in correspondence with , let satisfy condition (P2). If , then we get In any case, namely, Since is a positive linear operator, by applying to (22), for every and , we get and, hence, Let . By applying the modular , from (24) for any , we get So, to prove the theorem, it is sufficient to find a positive real number , with . Indeed, let be with for each : such a , by hypothesis, does exist. Choose an with for each and , and let be with . Taking into account (P1), for every , we get So, . Moreover, thanks to the choice of and , it is not difficult to see that .
As is -quasi-semiconvex and , then By taking the limit superior and thanks to -, from (25) and (27), we obtain From (28), by arbitrariness of and strong finiteness of , we get and hence . Thus, the double sequence , , is -modularly convergent to in .
The proof of the last part of the theorem is analogous.

We now give the main Korovkin-type theorem.

Theorem 6. Let be a monotone, strongly finite, absolutely continuous, and -quasi-semiconvex modular on and , , a double sequence of positive linear operators fulfilling -. If is -strongly convergent to , , in , then is -modularly convergent to in for every with , where and are as before.

Proof. Let with . By Proposition 4, there are a and a sequence , , in with and in the usual sense.
Fix arbitrarily and pick a positive integer with For every , we have By virtue of Theorem 5, we get By property -, we find an with From (30), (31), (32), (33), and subadditivity of the , we obtain From (34) and arbitrariness of , it follows that and so , that is, the assertion.

Remark 7. (a) Note that, in Theorem 6, in general, it is not possible to obtain -strong convergence unless the modular satisfies the condition; namely, there is a positive real number with for any (for the classical case, see also [48]).
(b) By a similar technique, we can prove an analogous result in the space of all continuous real-valued functions on of period , by the homeomorphic identification of with .

4. Rates of --Statistical Convergence

We present some estimates of rates of triangular -statistical convergence for Korovkin-type theorems in the setting of modular convergence, extending earlier results proved in [32].

Definition 8. Let be a nonnegative regular summability matrix and a nonincreasing sequence of positive real numbers. A double sequence is --statistically convergent to a number with the rate of if, for every , where In this case, we write

Definition 9. Let and be as in Definition 8. Then, a double sequence is --statistically bounded with the rate of if there is an with where In this case, we write

We now recall the following results (see also [32, Lemmas 3 and 4]).

Lemma 10. Let and be two double sequences, a nonnegative regular summability matrix, and and two positive nonincreasing sequences. If and as , then we have(i) as , where for each ,(ii) as for any real number .Furthermore, similar results hold when the symbol is replaced by .

Now, we are in position to prove the following.

Theorem 11. Let , , and be as above, let be a monotone and strongly finite modular, and suppose that is a metric space satisfying condition (). Let and be two nonincreasing sequences of strictly positive real numbers, and put for each . For each , set , and for every and let , where the symbol denotes the sup-norm and the supremum is taken with respect to the support of . Furthermore, let there exist with(i) as ,(ii) as .
Then for every we get as .
Furthermore, similar results hold when the symbol is replaced by .

Proof. Let , . Observe that for each . Using the properties of the modulus of continuity, we get for every and . Moreover, observe that, since , the quantities , , are well-defined. Indeed the support of is (totally) bounded, and so . By applying , , we find a positive real number (depending on , , and , but not on and ) with . From this, it follows that for each .
Let now . By applying , keeping fixed and letting vary in , and taking into account linearity and monotonicity, from (42), we obtain for each . Let now . By applying the modular , from (43), we obtain For each and , set From (44), it follows that and hence Letting tend to , we obtain that is, the assertion.

5. An Extension to Nonpositive Operators

One can ask whether it is possible in the Korovkin-type theorems to relax the positivity condition on the linear operators involved. We now give some positive answers also in the context of --statistical convergence, extending earlier results of  [12, 25], proved in the settings of -convergence and filter convergence, respectively.

Let be a nonnegative regular matrix, a bounded interval of , and (resp., ) the space of all functions defined on (resp., bounded and) continuous together with their first and second derivatives, .

Let , , and , , be functions in and let , , be as in (19), and suppose that satisfies the above conditions (P1) and (P2) and the following condition:(P3)there is a positive real constant with for all (here the second derivative is intended with respect to ).

Some examples in which properties (P1), (P2), and (P3) are satisfied can be found in [36].

We now prove the following Korovkin-type theorem for not necessarily positive linear operators, in the setting of --statistical convergence.

Theorem 12. Let be a nonnegative regular summability matrix, let be as in Theorem 5, and let , , , and , , satisfy properties (P1), (P2), and (P3). Assume that , , is a double sequence of linear operators, satisfying property -, and that . If is -modularly convergent to , , in , then is -modularly convergent to in for each .
If is -strongly convergent to , , in , then is -strongly convergent to in for every .
Furthermore, if is absolutely continuous and is -strongly convergent to , , in , then is -modularly convergent to in for every with .

Proof. Let . Note that is uniformly continuous and bounded on . Fix arbitrarily . By uniform continuity of there is a with for all , .
Let , , be as in (19), and let be associated with , satisfying (P2). By arguing analogously as in the proof of Theorem 5, for every and , we get where . From (49), it follows that for all and . Let satisfy (P3). For each , we have Since is bounded on , we can choose in such a way that and for each . Thus, . Let : by hypothesis, and From (50), (52), and linearity of , for each and , we have and, hence, By arguing analogously as in the proof of Theorem 5, using the modular and taking into account that , we get the assertion of the first part. The other parts can be proved by proceeding similarly as in the proofs of Theorems 5 and 6.

6. Some Application to Mellin-Type Operators

In this section, we give some application of Theorem 6 to bivariate moment-type operators (see also [37, 52]). We will consider the case of an Orlicz space , with and .

For every , let , , , , , and . For each , set , For every and , define . Moreover, let be the Cesàro matrix. Finally, let us consider the double sequences of operators defined by We will prove that the ’s satisfy Theorem 6. We begin with the following.

Lemma 13 (see [32, Lemma 1]). For every , we get where is as in (55).

Lemma 14 (see also [32, Lemma 2]). For every , there exists with for every , where denotes the ball with center and radius .

Lemma 15. For every , one gets .

Proof. Let . Then, taking into account Lemma 13, we have

Lemma 16. For each and , one gets .

Proof. By virtue of the Jensen inequality and the Fubini theorem, we have We now show that the hypotheses of Theorem 6 are satisfied. First of all, note that, by construction, we get for each and .
Fix now and let be the ball of center and radius . For every , , and , we get From this and arbitrariness of , it follows that for and for every there exists with Now, fix arbitrarily . From (62), since and taking into account convexity of the function , we get From (63) and Remark 2(b), it follows that , . Arguing analogously as above, considering for the estimate and taking into account Remark 2(b), it is possible to check that and so, by linearity, we get . Furthermore, note that, from Lemma 16, it follows that the condition - is satisfied, with (see also [14, Section 3]). Thus, by virtue of Theorem 6, for every , there is with .

We now consider a direct extension to the bivariate case of the classical one-dimensional moment kernel (see also [37, 52]).

For every and , let , and for and set We have and so for all and . Furthermore, we get