RATE OF CONVERGENCE OF STANCU BETA OPERATORS FOR FUNCTIONS OF BOUNDED VARIATION

. In this paper we study beta operators of second kind recently introduced by Prof. D. D. Stancu. We obtain an estimate on the rate of convergence for functions of bounded variation by means of the decomposition technique.


INTRODUCTION
In order to approximate Lebesgue integrable functions on the interval I = (0, ∞), in 1995 D. D. Stancu [4] defined beta operators L n of second kind given by (1) ( (1+t) nx+n+1 f (t) dt and investigated their approximation properties.Obviously the operators L n are positive linear operators on the space of locally integrable functions on I of polynomial growth as t → ∞, provided that n is sufficiently large.The fact that the operators (1) preserve linear functions is advantageous for their approximation properties.Recently Abel [1] derived the complete asymptotic expansion for the sequence of these operators.
In the present paper we study the rate of convergence of the operators L n by deriving an estimate of (L n f ) (x) − 1 2 {f (x+) + f (x−)} for functions f of bounded variation (see Theorem 1).
For the sake of a convenient notation in the proofs we rewrite the operators (1) as where the kernel function K n is given by Throughout this note, for fixed x ∈ I, we use the auxiliary function f x , which is defined by The following theorem is our main result.
Theorem 1.Let r > 0 and let f be a function of bounded variation on each finite subinterval of I satisfying f (t) = O (t r ) , as t → ∞.Fix a point x ∈ I. Then for each ε > 0 there exists an integer n (ε), such that for all n ≥ n (ε) there holds The proof of Theorem 1 is contained in Section 3, while the next section contains some auxiliary results used in the proof.

AUXILIARY RESULTS
In this section we give several results, which are necessary to prove Theorem 1.
For fixed x ∈ R, define the function ψ x , by ψ x (t) = t − x.The first central moments for the operators L n are given by ( 5) ).In general we have the following result: Let fixed x ∈ I be fixed.For s = 0, 1, 2, . . .and n ∈ N, the central moments for the operators L n satisfy Lemma 3. Let x ∈ I and K n (x, t) be defined by Eq. (3).Then for n ≥ 2, we have and Proof.First, we prove (i).In view of Eq. ( 5), we have The proof of (ii) is similar.
Lemma 4. For each x > 0, we have Proof.Let x > 0, and put By a change of variable we obtain and h x (1/x) = 0, h x is strictly decreasing on (1/x, ∞).Furthermore, , and Thus it is well-known (e.g., I x (n) meets the assumptions of [2, Theorem 1, Kap. 3, §5]), that there holds the complete asymptotic expansion .
By direct calculation we obtain the explicit expressions Thus we have On the other hand, application of Stirling's formula yields . Combination of both asymptotic expressions proves Lemma 4.

PROOF OF THE MAIN THEOREM
We close the paper by giving the proof of the main theorem.
Proof of Theorem 1.Let x ∈ I. Our starting-point is the inequality , where sign x is defined by sign x (t) = sign (t − x).
In order to prove the theorem we need estimates for (L n f x ) (x) and (L n sign x ) (x).We first estimate (L n sign x ) (x) as follows: With the notation (2) we obtain By Lemma 4, for each ε > 0 there exists an integer n = n (ε), such that Next we estimate L n (f x , x) as follows: We first estimate E 1 .Writing y = x − x/ √ n and we have, integrating by parts, By using (i) of Lemma 3, we get Integrating by parts the last term we have Now replacing the variable y in the last integral by x − x/ √ n, we obtain and therefore, (9) Finally, we estimate E 3 .We put and divide E 3 = E 31 + E 32 , where With y = x + x/ √ n the first integral can be written in the form By Eq. (ii) of Lemma 3, we conclude, for sufficiently large n, Using the similar method as above we obtain 2x y which implies the estimate Lastly, we estimate E 32 .By assumption, there exists an integer r, such that f (t) = O t 2r , as t → ∞.Thus, for a certain constant C > 0 depending only on f , x and r, we have Collecting the estimates (9)-( 12), we have, by Eq. ( 8), Combining the estimates of (6), (7) and (13) completes the proof of the theorem.