Application of f-lacunary statistical convergence to approximation theorems

The concept of f-lacunary statistical convergence which is, in fact, a generalization of lacunary statistical convergence, has been introduced recently by Bhardwaj and Dhawan (Abstr. Appl. Anal. 2016:9365037, 2016). The main object of this paper is to prove Korovkin type approximation theorems using the notion of f-lacunary statistical convergence. A relationship between the newly established Korovkin type approximation theorems via f-lacunary statistical convergence, the classical Korovkin theorems and their lacunary statistical analogs has been studied. A new concept of f-lacunary statistical convergence of degree β (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0 < \beta< 1$\end{document}0<β<1) has also been introduced, and as an application a corresponding Korovkin type theorem is established.


Density by moduli and statistical convergence
The idea of statistical convergence, which is, in fact, a generalization of the usual notion of convergence, was first introduced by Fast [14] and Steinhaus [37] independently in 1951 and since then several generalizations and applications of this concept have been investigated by various authors, namelySalát [35], Fridy [16], Aizpuru et al. [1], Aktuğlu [2], Gadjiev and Orhan [18], Mursaleen and Alotaibi [28], and many others.
Statistical convergence depends on the natural density of subsets of the set N = {1, 2, 3, . . .}. The natural density d(K) of a set K ⊆ N (see [29,Chap. 11]) is defined by where |{k ≤ n : k ∈ K}| denotes the number of elements of K not exceeding n. Obviously, we have d(K) = 0 if K is finite.
Definition 1.1 ([22]) Let X be a normed space. A sequence (x k ) in X is said to be statistically convergent to some x ∈ X, if for each > 0 the set {k ∈ N : x kx ≥ } has natural density zero, i.e., Definition 1.5 Let θ = (k r ) be a lacunary sequence. A number sequence (x k ) is said to be lacunary statistically convergent to l, or S θ -convergent to l, if for each ε > 0, lim r→∞ 1 h r k ∈ I r : |x k -l| ≥ ε = 0.
In this case, we write S θ -lim x k = l.
Quite recently, Bhardwaj and Dhawan [6] have extended the concept of lacunary statistical convergence to that of f -lacunary statistical convergence as follows: Definition 1. 6 Let f be an unbounded modulus and θ = (k r ) be a lacunary sequence. A number sequence (x k ) is said to be f -lacunary statistically convergent to l, or S f θconvergent to l, if for each ε > 0, In this case, we write S f θ -lim x k = l.
An extension of the concepts of lacunary statistical convergence and f -lacunary statistical convergence in a more general setting of normed spaces shall be needed in the present work and is given below. Definition 1.7 Let X be a normed space and θ = (k r ) a lacunary sequence. A sequence (x k ) in X is said to be lacunary statistically convergent to x ∈ X, if, for each ε > 0, lim r→∞ 1 h r k ∈ I r : x kx ≥ ε = 0.
In this case, we write S θ (X) -lim x k = x. However, if there is no confusion regarding the scalar-or vector-valued sequences, we may avoid writing X explicitly, i.e., we may simply write S θ -lim x k = x. The set of all X-valued lacunary statistically convergent sequences is denoted by S θ (X).

Definition 1.8
Let f be an unbounded modulus, X a normed space, and θ = (k r ) a lacunary sequence. A sequence (x k ) in X is said to be f -lacunary statistically convergent to x ∈ X, if, for each ε > 0, In this case, we write S f θ (X) -lim x k = x, or simply S f θ -lim x k = x, as mentioned above. The set of all X-valued f -lacunary statistically convergent sequences is denoted by S f θ (X).

Korovkin-type approximation theorems
The theory of approximation is an area of mathematical analysis, which, at its core, is concerned with the approximation of functions by simpler and more easily calculated functions. In the 1950s, the theory of approximation of functions by positive linear operators developed a lot, when Popoviciu [33], Bohman [11] and Korvokin [23,24] independently discovered a simple and easily applicable criterion to check if a sequence of positive linear operators converges uniformly to the function to be approximated. This criterion says that the necessary and sufficient condition for the uniform convergence of the sequence (L n ) of positive linear operators to the continuous function g on the compact interval [a, b] is the uniform convergence of the sequence (L n g) to g for only the three functions e n (x) = x n , n = 0, 1, 2. This classical result of approximation theory is mostly known under the name of Bohman-Korovkin theorem, because Popoviciu's contribution in [33] remained unknown for a long time.
Due to this classical result, the monomials e n , n = 0, 1, 2, play an important role in the approximation theory of linear and positive operators on spaces of continuous functions. These monomials are often called Korovkin test-functions. This elegant and simple result has inspired many mathematicians to extend this result in different directions, generalizing the notion of sequence and considering different spaces. In this way a special branch of approximation theory arose, called Korovkin-type approximation theory. A complete and comprehensive exposure on this topic can be found in [3].
Statistical convergence had not been examined in approximation theory until 2002. Korovkin first and second approximation theorems were first proved via statistical convergence by Gadjiev and Orhan [18] and Duman [13], in 2002 and 2003, respectively. In 2005, Patterson and Savaş [30] proved the first Korovkin approximation theorem via lacunary statistical convergence. It is quite interesting to note that the lacunary statistical analog of the Korovkin second approximation theorem has not been studied so far. Korovkin-type approximation theorems have been studied via various summability methods by many mathematicians. Quite recently Bhardwaj and Dhawan [8] have obtained fstatistical analogs of the classical Korovkin first and second approximation theorems. For a detailed account one may refer to [2,4,12,20,21,27,28] where many more references can be found.

Correct reformulation of the various analogs of the classical Korovkin first theorem
The authors wish to thank Professor F. Altomare for his help in the correct reformulation of the various analogs of the classical Korovkin first theorem. For a given closed and bounded interval [a, b], we first introduce the following spaces: We also recall [32] that for any linear spaces X, Y of real functions 1. The mapping L : X → Y is called a linear operator if L(αf + βg) = αL(f ) + βL(g) for f , g ∈ X and α, β ∈ R.
2. If f ≥ 0, f ∈ X ⇒ Lf ≥ 0, then L is a positive linear operator. 3. In order to highlight the argument of the function Lf ∈ Y , we use the notation L(f , x). The classical Korovkin first theorem [24] is stated in Gadjiev and Orhan [18] as follows:   [18] is essentially the same as the space F cb ([a, b]) defined above. A new notation for the same space has been introduced for the sake of notational uniformity. We are now ready to give the correct reformulation of Theorem 1.9 as follows: In the same paper [18], Gadjiev Patterson and Savaş [30] have given the lacunary statistical analog of Korovkin first theorem as follows: Remark 1.17 The same inaccuracy gets repeated here also. The corrected version is as follows: We conclude this section by stating the recently obtained (see [8]) f -statistical analogs of the Korovkin first and second approximation theorems as we shall be needing them later in this paper.

Discussion of the main problem
In this paper we mainly prove Korovkin-type approximation theorems via f -lacunary statistical convergence. The lacunary statistical analog of the Korovkin second approximation theorem is obtained as a particular case. A relationship between the newly established Korovkin type approximation theorems via f -lacunary statistical convergence, the classical Korovkin theorems and their lacunary statistical analogs has been studied. In addition, we also establish a relationship between the f -lacunary statistical analogs and f -statistical analogs of classical Korovkin first and second approximation theorems. The proofs of our main results, i.e., Theorems 2.3 and 2.13, may appear to contain same calculations from the corresponding old ones but, in fact, there are certain gaps and mistakes in the corresponding earlier published proofs which have been filled in and corrected.

f -lacunary statistical analog of the Korovkin first theorem
In order to prove an f -lacunary statistical analog of Korovkin first theorem, we need the following lemma.
Remark 2.2 In the above lemma, there is an inaccuracy in the sense that, when Korovkin assumes that "f is continuous in the interval [a, b]", then this means for him that f |[a, b] is continuous. For these reasons he adds the additional hypotheses that f is continuous on the right at the point b and that f is continuous on the left at the point a. According to the modern terminology, when we assume that f ∈ F c ([a, b]), then f is continuous at every point of [a, b] and, hence, in particular at a (both on the right and on the left) as well as at b (both on the right and on the left). Therefore, in the statement of Theorem 1.13 above, due to Gadjiev and Orhan [18], it is correctly stated that formula st -lim L n (g, x)g(x) B = 0 holds for every g ∈ C M [a, b] (i.e., F cb ([a, b])) because, for such functions Lemma 2.1 can be applied.
We are now in a position to state and prove the promised f -lacunary statistical analog of the Korovkin first theorem.
Thus, for any > 0, we have and, using (2.14), get Remark 2.4 Since every convergent sequence is f -lacunary statistically convergent [6], it immediately follows that any sequence satisfying the conditions of the classical Korovkin first theorem automatically satisfies the conditions of its f -lacunary statistical analog.
Our next example shows that there may exist a sequence of positive linear operators which satisfies the conditions of Theorem 2.3 but does not satisfy the conditions of Theorem 1.12, thereby showing that our result is stronger than the classical one. and (α n ) is the sequence of scalars which is f -lacunary statistically convergent to zero for some unbounded modulus f but not convergent to zero. Before proceeding further, we give a specific example of such type of a sequence (α n ) as follows.
Let f be an unbounded modulus, for which lim t→∞ t > 0 and there is a positive constant c such that f (xy) ≥ cf (x)f (y) for all x ≥ 0, y ≥ 0. Proceeding as in [15, p. 511], let θ = (k r ) be a lacunary sequence and (k r(j) ) a subsequence of lacunary sequence θ such that q r(j) > j. Define a bounded sequence α = (α k ) by ple of a sequence of scalars which is f -lacunary statistically convergent to zero for some unbounded modulus f but not convergent to zero. It is known (see [24]) that Hence We next claim that the lacunary statistical analog of the classical Korovkin first theorem is stronger than the f -lacunary statistical analog of the classical Korovkin first theorem. For this we first provide an example of a lacunary statistically convergent sequence which is not f -lacunary statistically convergent.
For 0 < ε ≤ 1, by construction of the sequence x = (x k ), we have Therefore, Now consider the unbounded modulus function f (x) = log(1 + x). We will show that S f θ -lim x k = 0, whence it will follow that (x k ) is not f -lacunary statistically convergent. Indeed, suppose (x k ) were f -lacunary statistically convergent to some number l, then by Theorem 11 of [6], (x k ) would be lacunary statically convergent to l and, finally, by the uniqueness of S θ -limit for a fixed θ (see [17], page 48), this l had to be 0.
Our next example shows that there exists a sequence of positive linear operators which satisfies the conditions of Theorem 1.18 but does not satisfy the conditions of Theorem 2.3, thereby implying that the lacunary statistical analog of the classical Korovkin first theorem is stronger than the f -lacunary statistical analog of the classical Korovkin first theorem.
where (B n ) is the sequence of classical Bernstein polynomials and (α n ) is any sequence of scalars which is lacunary statistically convergent to zero but not f -lacunary statistically convergent to zero for some unbounded modulus f . It is easy to see, as in Example 2.5, that the sequence (Q n ) satisfies the lacunary statistical analog of Korovkin first theorem but does not satisfy the f -lacunary statistical analog of Korovkin first theorem.
We now study a relationship between the f -lacunary statistical analog and the fstatistical analog of the Korovkin first theorem. In other words, we characterize those θ for which these two analogs become equivalent, of course, under certain restrictions on f . In order to do this, we need the following lemmas which are actually simple extensions of Lemmas 17 and 19 of Bhardwaj and Dhawan [6] to an arbitrary normed space. Combining Lemmas 2.9 and 2.10, we have the following.  Then, the f -lacunary statistical analog and f -statistical analog of the Korovkin first theorem are equivalent for those θ for which 1 < lim inf r q r ≤ lim sup r q r < ∞.

f -lacunary statistical analog of Korovkin second theorem
The classical Korovkin second theorem [24] may be stated as follows. We now prove an f -lacunary statistical analog of the Korovkin second theorem, from which the lacunary statistical analog is obtained as a particular case.
Thus inequality (2.21), on which the proof of this theorem is based, has not yet been established for all t, but it holds for t belonging to the subinterval (xδ, 2π + xδ] of length 2π . In order to prove validity of inequality (2.21) for all t ∈ (-∞, ∞), we note that the function has period 2π and, according to the conditions of the theorem, the function g (t) also has period 2π , i.e., ψ(t + 2kπ) = ψ(t) and g (t + 2kπ) = g (t).
Using inequality (2.21) and monotonicity of the operator L n (g , x), we obtain As we know x is fixed and so g (x) is a constant number. Therefore, (2.23) Also, Substituting the value of L n (ψ, x) in (2.25), we get For any > 0, choose > 0 such that < . Now, from inequality (2.26), we have Then it is easy to see that D ⊂ D 1 ∪ D 2 ∪ D 3 . Now, from (2.27), we have which yields and, using (2.16)-(2.18)), we get From here onwards, we proceed as in the proof of Theorem 2.3 to get Remark 2. 15 If we take f (x) = x in Theorem 2.14, we obtain the lacunary statistical analog of the classical Korovkin second theorem as follows.
Remark 2.17 Since every convergent sequence is f -lacunary statistically convergent [6], it immediately follows that any sequence satisfying the conditions of the classical Korovkin second theorem automatically satisfies the conditions of its f -lacunary statistical analog.
Our next example shows that there exists a sequence of positive linear operators which satisfies the conditions of Theorem 2.14 but does not satisfy the conditions of Theorem 2.13, thereby showing that our result is stronger than the classical one.
Example 2.18 Following Duman [13], consider the sequence Q n : D * ([-π, π]) → B([-π, π]) of positive linear operators defined by where (F n ) is the sequence of Fejer operators defined by and (α n ) is the sequence of scalars which is f -lacunary statistically convergent to zero for some unbounded modulus f but not convergent to zero. It is known [24] that F n (1, x) = 1, F n (cos t, x) = n -1 n cos x and F n (sin t, x) = n -1 n sin x.
Hence, the sequence (Q n ) satisfies conditions (2.16)-(2.18) of Theorem 2.14. So, we have On the other hand, and so, lim Q n (1, x) -1 B = lim 1 + α n -1 = lim α n = lim |α n | = 0, from where it follows that (Q n ) does not satisfy the conditions of classical Korovkin second theorem.
Remark 2.19 Since every f -lacunary statistically convergent sequence is lacunary statistically convergent [6], it immediately follows that any sequence satisfying the conditions of the f -lacunary statistical analog of the classical Korovkin second theorem (Theorem 2.14) automatically satisfies the conditions of the lacunary statistical analog of the classical Korovkin second theorem (Theorem 2.16).
Remark 2.20 In Example 2.18, if we take (α n ) to be any sequence which is lacunary statistically convergent to zero but not f -lacunary statistically convergent to zero for some unbounded modulus f , then we obtain a sequence of positive linear operators which satisfies the conditions of Theorem 2.16 but does not satisfy the conditions of Theorem 2.14, thereby showing that the lacunary statistical analog of the classical Korovkin second theorem is stronger than the f -lacunary statistical analog of the classical Korovkin second theorem.
We conclude this section by studying a relationship between the f -lacunary statistical analog and the f -statistical analog of the Korovkin second theorem. In other words, we characterize those θ for which these two analogs are equivalent, of course, under certain restrictions on f . In view of Theorems 1.20, 2.11 and 2.14, we have the following.

Theorem 2.21
Let f be any unbounded modulus, for which lim t→∞ f (t) t > 0 and there is a positive constant c such that f (xy) ≥ cf (x)f (y) for all x ≥ 0, y ≥ 0. Then, the f -lacunary statistical analog and f -statistical analog of the Korovkin second theorem are equivalent for those θ for which 1 < lim inf r q r ≤ lim sup r q r < ∞.

The order of f -lacunary statistical convergence
The idea of lacunary statistical convergence with degree β (0 < β < 1) for sequences of numbers was introduced by Patterson and Savaş [30] as follows:

Definition 2.22
The number sequence (x k ) is said to be lacunary statistically convergent to the number l with degree 0 < β < 1 if, for each > 0, In this case, we write The concept of lacunary statistical convergence of order α was introduced by Şengül and Et [36] as follows:

Conclusion
New versions of Korovkin type approximation theorems using the notion of f -lacunary statistical convergence have been established. It is shown that any sequence satisfying the conditions of the classical Korovkin first (second) theorem satisfies the conditions of its corresponding f -lacunary statistical analog whereas there exists a sequence of positive linear operators which satisfies the conditions of f -lacunary statistical analog of Korovkin first (second) theorem without satisfying the conditions of the corresponding classical Korovkin theorem, thereby showing that our results are stronger than the classical ones.
We have also shown that lacunary statistical analog of Korovkin first (second) theorem is stronger than the f -lacunary statistical analog of Korovkin first (second) theorem.
Finally, we have characterized those θ for which f -lacunary statistical analog and the f -statistical analog of the Korovkin first (second) theorem are equivalent, of course, under certain restrictions on f .