Fibonacci statistical convergence and Korovkin type approximation theorems

The purpose of this paper is twofold. First, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined. Second, we provide various approximation results concerning the classical Korovkin theorem via Fibonacci type statistical convergence.

Let f be any density. Then, for any set of natural numbers A, the function f is said to be upper density associated with f if f (A) = f (Z + \A).
Consider the set A ⊂ Z + . If f (A) = f (A), then we can say that the set A has natural density with respect to f . The term asymptotic density is often used for the function where A ⊂ N and A(n) = a≤n,a∈A . Also the natural density of A is given by d(A) = lim n n - |A(n)|, where |A(n)| denotes the number of elements in A(n).
The study of statistical convergence was initiated by Fast []. Schoenberg [] studied statistical convergence as a summability method and listed some of the elementary properties of statistical convergence. Both of these mathematicians mentioned that if a bounded sequence is statistically convergent to L, then it is Cesàro summable to L. Statistical convergence also arises as an example of 'convergence in density' as introduced by Buck  Definition . A real numbers sequence x = (x k ) is statistically convergent to L provided that for every ε >  the set {n ∈ N : |x n -L| ≥ ε} has natural density zero. The set of all statistically convergent sequences is denoted by S. In this case, we write S -lim x = L or x k → L(S).

Definition . ([])
The sequence x = (x k ) is statistically Cauchy sequence if for every ε >  there is a positive integer N = N(ε) such that It can be seen from the definition that statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence.
Fridy [] introduced a new notation for facilitation: If x = (x n ) is a sequence that satisfies some property P for all n except a set of natural density zero, then we say that x = (x n ) satisfies P for 'almost all n' , and we abbreviate 'a.a.n' . In [], Fridy proved the following theorem.
Theorem . The following statements are equivalent: i. x is a statistically convergent sequence; ii. x is a statistically Cauchy sequence; iii. x is a sequence for which there is a convergent sequence y such that x n = y n for a.a.n.

Fibonacci numbers and Fibonacci matrix
The numbers in the bottom row are called Fibonacci numbers, and the number sequence , , , , , , , , , , , , . . .
Definition . The Fibonacci numbers are a sequence of numbers (f n ) for n = , , . . . defined by the linear recurrence equation From this definition, it means that the first two numbers in Fibonacci sequence are either  and  (or  and ) depending on the chosen starting point of the sequence and all subsequent numbers is the sum of the previous two. That is, we can choose Fibonacci sequence was initiated in the book Liber Abaci of Fibonacci which was written in . However, the sequence is based on older history. The sequence had been described earlier as Virahanka numbers in Indian mathematics []. In Liber Abaci, the sequence starts with , nowadays the sequence begins either with f  =  or with f  = .
Some of the fundamental properties of Fibonacci numbers are given as follows: .
can substitute for f n+ in Cassini's formula. Let f n be the nth Fibonacci number for every n ∈ N. Then we define the infinite matrix (≤ k < n - or k > n). Korovkin

Methods
In the theory of numbers, there are many different definitions of density. It is well known that the most popular of these definitions is asymptotic density. However, asymptotic density does not exist for all sequences. New densities have been defined to fill those gaps and to serve different purposes.
The asymptotic density is one of the possibilities to measure how large a subset of the set of natural numbers is. We know intuitively that positive integers are much more than perfect squares. Because every perfect square is positive and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless, if one goes through the natural numbers, the squares become increasingly scarce. It is precisely in this case that natural density helps us and makes this intuition precise.
The Fibonacci sequence was firstly used in the theory of sequence spaces by Kara  In this paper, by combining the definitions of Fibonacci sequence and statistical convergence, we obtain a new concept of statistical convergence, which will be called Fibonacci type statistical convergence. We examine some basic properties of new statistical convergence defined by Fibonacci sequences. Henceforth, we get an analogue of the classical Korovkin theorem by using the concept of Fibonacci type statistical convergence.
It will be shown that if X is a Banach space, then for a closed subset of X, which is denoted by A, Fibonacci type space A is closed in Fibonacci type space X. We will give the definitions of Fibonacci statistically Cauchy sequence and investigate the Fibonacci statistically convergent sequences and Fibonacci statistically Cauchy sequences. Using the definition of statistical boundedness, it will be proved that the set of Fibonacci statistically convergent sequence spaces of real numbers is a closed linear space of a set of Fibonacci bounded sequences of real numbers and nowhere dense in Fibonacci bounded sequences of real numbers. After proving that the set of Fibonacci statistically convergent sequences is dense in Frechet metric space of all real sequences, the inclusion relations will be given.
For the rest of the paper, firstly an approximation theorem, which is an analogue of Korovkin theorem, is given and an example is solved. Second, the rate of Fibonacci statistical convergence of a sequence of positive linear operators defined C π (R) into C π (R) is computed.

Fibonacci type statistical convergence
Now, we give the general Fibonacci sequence space X( F) as follows [, ]: Let X be any sequence space and k ∈ N. Then It is clear that if X is a linear space, then X( F) is also a linear space. Kara proved that if X is a Banach space, then X( F) is also a Banach space with the norm Now, we will give lemma which is used in the proof of Theorem .. Proof of this lemma is trivial.

Theorem . Consider that X is a Banach space and A is a closed subset of X. Then A( F) is also closed in X( F).
Proof Since A is a closed subset of X, from Lemma ., then we can write A( F) ⊂ X( F). A( F), A denote the closure of A( F) and A, respectively. To prove the theorem, we must show that A( F) = A( F).
Firstly, we take x ∈ A( F). Therefore, from .. Theorem of [], there exists a sequence ( From this theorem, we can give the following corollary. In this case we write d( F) -lim x k = L or x k → L(S( F)). The set of F-statistically convergent sequences will be denoted by S( F). In the case L = , we will write S  ( F).
Theorem . If x is an F-statistically convergent sequence, then x is an F-statistically Cauchy sequence. Proof Suppose that Fx k = Fy k for almost all k and y k → L(S( F)). Then ε >  and, for each Since y k → L(S( F)), the latter set contains a fixed number of integers, say g = g(ε). Therefore, for Fx k = Fy k , for almost all k, By m  , we denote the linear space of all statistically bounded sequences. Bounded sequences are obviously statistically bounded as the empty set has zero natural density. However, the converse is not true. For example, we consider the sequence Clearly (x k ) is not a bounded sequence. However, d({k : |x k | > /}) = , as the set of squares has zero natural density and hence (x k ) is statistically bounded [].

Proposition . ([]) Every convergent sequence is statistically bounded.
Although a statistically convergent sequence does not need to be bounded (cf. [, ]), the following proposition shows that every statistically convergent sequence is statistically bounded.

Proposition . ([]) Every statistically convergent sequence is statistically bounded.
Now, using Propositions . and ., we can give the following corollary.

Corollary . Every F-statistically convergent sequence is F-statistically bounded.
Denote the set of all F-bounded sequences of real numbers by m( F) []. Based on Definition . and the descriptions of m  and m( F), we can denote the set of all F-bounded statistically convergent sequences of real numbers by m  ( F).
The following theorem can be proved by Theorem . of [] and Theorem ..

Theorem . The set of m  ( F) is a nowhere dense set in m( F).
Proof According to [], every closed linear subspace of an arbitrary linear normed space E, different from E, is a nowhere dense set in E. Hence, on account of Theorem ., it suffices to prove that m  ( F) = m( F). But this is evident, consider the sequence

 (n is even).
Then x ∈ m( F), but does not belong to m  ( F).
ω denotes the Fréchet metric space of all real sequences with the metric d ω , where x = (x k ), y = (y k ) ∈ ω for all k = , , . . . .

Theorem . The set of F-statistically convergent sequences is dense in the space ω.
Proof If x = (x k ) ∈ S( F) (for all k) and the sequence y = (y k ) (for all k) of real numbers differs from x only in a finite number of terms, then evidently y ∈ S( F), too. From this statement the proof follows at once on the basis of the definition of the metric in ω. iii) The proof is the same as (ii). iv) Define Then x ∈ S, but x / ∈ S( F). Conversely, if we take u = (n), then u / ∈ S but x ∈ S( F).

.. Approximation by F-statistical convergence
In this section, we get an analogue of classical Korovkin theorem by using the concept of F-statistical convergence. Let F(R) denote the linear space of real-valued functions on R. Let C(R) be a space of all real-valued continuous functions f on R. It is well known that C(R) is a Banach space with the norm given as follows: and we denote by C π (R) the space of all π -periodic functions f ∈ C(R), which is a Banach space with the norm given by We say A is a positive operator if for every non-negative f and x ∈ I, we have A(f , x) ≥ , where I is any given interval on the real semi-axis. The first and second classical Korovkin approximation theorems state the following (see [, ]).

Theorem . Let (A n ) be a sequence of positive linear operators from C[, ] into F[, ]. Then
where e i = x i , i = , , .

Theorem . Let (T n ) be a sequence of positive linear operators from C π (R) into F(R). Then
Our main Korovkin type theorem is given as follows.
Theorem . Let (L k ) be a sequence of positive linear operators from C π (R) into C π (R).
Then, for all f ∈ C π (R), Proof As , sin x, cos x ∈ C π (R), conditions (.)-(.) follow immediately from (.). Let conditions (.)-(.) hold and I  = (a, a + π) be any subinterval of length π in R. Let us fix x ∈ I  . By the properties of function f , it follows that for given ε >  there exists δ = δ( ) >  such that If |t -x| ≥ δ, let us assume that t ∈ (x + δ, π + x + δ). Then we obtain that where ψ(t) = sin  ( t-x  ). By using (.) and (.), we have This implies that By using the positivity and linearity of {L k }, we get where x is fixed and so f (x) is a constant number. Therefore, On the other hand, we get By inequalities (.) and (.), we obtain Now, we compute the second moment By (.), we have So, from the above inequality, one can see that Because ε is arbitrary, we obtain ). Finally, replacing L k (·, x) by T k (·, x) = FL k (·, x) and for ε > , we can write . Thus, by conditions (.)-(.), we obtain which completes the proof.
We remark that our Theorem . is stronger than Theorem . as well as Theorem of Gadjiev and Orhan []. For this purpose, we get the following example.
Example . For n ∈ N, denote by S n (f ) the n-partial sum of the Fourier series of f , that is, For n ∈ N, we get A standard calculation gives that for every t ∈ R The sequence (ϕ n ) n∈N is a positive kernel which is called the Fejér kernel, and corresponding F n for n ≥  are called Fejér convolution operators. We define the sequence of linear operators as K n : . Then K n (, x) = , K n (sin t, x) = n n+ sin x and K n (cos t, x) = n n+ cos x and the sequence (K n ) satisfies conditions (.)-(.). Therefore, we get On the other hand, one can see that (K n ) does not satisfy Theorem . as well as Theorem of Gadjiev and Orhan [] since Fy = (, , , . . .), the sequence y is F-statistically convergent to . But the sequence y is neither convergent nor statistically convergent.

.. Rate of F-statistical convergence
In this section, we estimate the rate of F-statistical convergence of a sequence of positive linear operators defined by C π (R) into C π (R). Now, we give the following definition.
Definition . Let (a n ) be a positive non-increasing sequence. We say that the sequence x = (x k ) is F-statistically convergent to with the rate o(a n ) if, for every ε > , At this stage, we can write As usual we have the following auxiliary result.
Lemma . Let (a n ) and (b n ) be two positive non-increasing sequences. Let x = (x k ) and y = (y k ) be two sequences such that x k -L  = d( F)o(a n ) and y k - o(a n b n ), where c n = max{a n , b n }.
For δ > , the modulus of continuity of f , ω(f , δ) is defined by It is well known that for a function f ∈ C[a, b], Theorem . Let (L k ) be a sequence of positive linear operators from C π (R) into C π (R). Assume that Then, for all f ∈ C π (R), we get where z n = max{u n , v n }.

Conclusion
One of the most known and interesting number sequences is the Fibonacci sequence, and it still continues to be of interest to mathematicians because this sequence is an important and useful tool to expand the mathematical horizon for many mathematicians.
The concept of statistical convergence for a sequence of real numbers was defined by Fast [] and Steinhaus [] independently in . Statistical convergence has recently become an area of active research. Currently, researchers in statistical convergence have devoted their effort to statistical approximation.
Approximation theory has important applications in the theory of polynomial approximation in various areas of functional analysis. The study of the Korovkin type approximation theory is a well-established area of research, which is concerned with the problem of approximating a function f by means of a sequence A n of positive linear operators. Statistical convergence is quite effective in the approximation theory. In recent times, very high quality publications have been made using approximation theory and statistical convergence [, , -].
In this study, we have studied the concept of statistical convergence which has an important place in the literature using Fibonacci sequences. The statistical convergence is a generalization of the usual notion of convergence. We have defined the Fibonacci type statistical convergence and investigated basic properties. A new version of Korovkin type approximation theory was introduced using the new concept of statistical convergence.