Order of approximation by an operator involving biorthogonal polynomials

The goal of this paper is to estimate the rate of convergence of a linear positive operator involving Konhauser polynomials to bounded variation functions on [0,1]$[0,1]$. To prove our main result, we have used some methods and techniques of probability theory.


Introduction
In , Konhauser presented the general theory of biorthogonal polynomials []. Afterwards, in  [], he gave the following pair of biorthogonal polynomials: Y (n) ν (x; k) and Z (n) ν (x; k) (n > -, k ∈ Z + ), satisfying where L (n) ν (x) are classical Laguerre polynomials and Y (n) ν (x; k) Konhauser polynomials given by and In order to give the monotonicity properties, Cheney and Sharma introduced the following type modification of the MKZ operators: M * n (f ; ) = f ().
In [], they also introduced the operators for x ∈ [, ) and t ∈ (-∞, ] where L (n) k (t) denotes the classical Laguerre polynomials. We consider the sequence of linear positive operators (similarly to []), which is another generalization of these operators including where {F n (x, t)} n∈N are the generating functions for the sequence of functions and . If we choose k =  in (), then the operators turn out to be (). Similarly, if we choose k =  and t =  in (), we get (), which are called Bernstein power series by Cheney and Sharma in [].
In view of () and (), one has For simplicity, we set In view of (), we can write () as in which x ∈ [, ), t ∈ (-∞, ] and k < n + . Our paper concerns the rate of pointwise convergence of the operators given by (). In particular, by means of the techniques of probability theory, we shall estimate the rate of convergence for the operators (L n f ) for functions of a bounded variation on [, ] at points x where f (x+) and f (x-) exist.
It is worthwhile to say that our present results extend some earlier results. In fact, if we choose k =  in L n f , then the operators reduce to the operators mentioned [], and if k =  and t =  in L n f , then we get the operator investigated in [].
For some important papers on different operators related to the present study we refer the readers to Bojanic The main theorem of this work reads as follows.

Theorem . Let f be a function of a bounded variation on
. Then for each x ∈ (, ), t ∈ (-∞, ], k < n + , and n sufficiently large, we have

Some lemmas
We now mention certain results which are necessary to prove our main theorem.
Proof For the proof of () see [].
To prove (), we can use the following equality: From (), one has (). So, this completes the proof of Lemma ..
Taking derivatives of both sides with respect to x yields Editing the equation, we have The proofs of (), () are similar.

Lemma . Let ζ  be the random variable with
Then Proof For n = , we have If we use ()-(), the proof of the Lemma . is completed.
Next, we recall the well-known Berry-Esséen bound for the classical central limit theorem of probability theory.

Lemma . (Berry-Esséen) Let {ζ k } ∞ k= be a sequence of independent and identically distributed random variables with finite variance such that the expectation E(ζ
then there exists a constant C, / √ π ≤ C ≤ ., such that for all n = , , . . . and all t, Its proof can be found in Shiryayev [].
Proof By direct calculation, one has from Lemmas .-. the desired result.

Lemma . For all x ∈ (, ) define the function sgn(sx) by
We have Proof If we apply the operator L n to the function of sgn(sx), we have and we can write where δ x (s) is the Dirac delta function, C * is a certain constant, and A(x, t; k) is given in Theorem ..
Proof By direct calculation, we get One has According to Lemma ., one has and using the method of proof of Lemma . and evaluations which are similar to the work in [], we have m n x, t; k =: Set C * = max{C, C  }. Consequently from () and () we get (). This completes the proof of Lemma ..
Lemma . For n sufficiently large, we have Proof We recall the Lebesgue-Stieltjes integral representations where λ n (x, t, y; k) = y  K n (x, t, s; k) ds,  ≤ s ≤ x. From (), we can rewrite (L n g x )(x, t; k) as follows: To estimate (), we decompose it into three parts as follows: where x-x/ √ n g x (y) d y λ n (x, t, y; k), We shall evaluate (I  g x )(x, t), (I  g x )(x, t), and (I  g x )(x, t).
First we estimate ( () Next we estimate (I  g x )(x, t). Using partial Lebesgue-Stieltjes integrations, we obtain Since |g From Lemma ., it is clear that It follows that Furthermore, since and putting y = xx/ √ u in the last integral, we get Consequently,