APPROXIMATION PROPERTIES OF MODIFIED STANCU BETA OPERATORS

. In this paper we give approximation theorems for modiﬁed Stancu beta operators of diﬀerentiable functions. The Stancu beta operators were examined in [8, 1, 2, 5] and other papers. MSC 2000. 41A36, 41A25.

In [8] it was proved that if f is continuous and bounded on I, then , for all x ∈ I and n ≥ 2, where ω k (f ; •), k = 1, 2, is the modulus of continuity of the order k of f .
It is known ( [1], [8]) that if f is continuous and bounded on I with derivatives f and f , then (1.5) lim 2 f (x) at every x ∈ I and the order O 1 n of approximation of r times differentiable functions f , r ≥ 3, by L n cannot be improved.

1.2.
In this paper we shall show that the approximation order of differentiable functions by beta operators can be improved by some modification of formula (1.1).We use the Kirov type method given for Bernstein operators in [6] (see also [7]).
Similarly to [3] and let C p be the set of all real-valued functions f defined on I, for which w p f is uniformly continuous and bounded on I and the norm (1.7) We have In this paper we shall consider the functions class

1.3.
Analogously to [8] we denote by for positive parameters u and v, where B is the beta function defined by (1.2).Let r ∈ N 0 .For f ∈ C r we define the following modified Stancu beta operators: (1.9) x ∈ I and r ≤ n ∈ N , where If r = 0 and f ∈ C 0 , then by (1.9), (1.10) and (1.1) we have In Section 2 we shall prove that L n;r , r ∈ N , n ≥ 2r, is a linear operator acting from C r into C r .Moreover we shall prove that L n is a linear positive operator acting from C r into C r if n ≥ 2r and r ∈ N .

1.4.
It is known ( [1], [8]) that operators L n (f ) given by (1.1) are well defined for functions f s (x) = x s , x ∈ I, s ∈ N , n ≥ s, and L n (f s ) are algebraic polynomials of the order s.In [1] and [8] it was proved that (1.12) and for every s ∈ N there exists a positive constant M 1 (s) depending only on s such that

LEMMAS
In this paper we shall denote by M i (α, β), i ∈ N , suitable positive constants depending only on indicated parameters α and β.
We shall apply the following inequalities for x > 0 and p, s ∈ N 0 , which can be easily obtained from (1.6).
Proof.By (1.6) we have which by (1.1), (1.2) and (1.12) implies that and we complete the proof.
Lemma 2.3.Let p ∈ N .Then there exists M 5 (p) = const.> 0 such that for every f ∈ C p we have for every f ∈ C r and n ≥ 2r.

3.1.
First we shall give theorems on the order of approximation of function f by operators L n and L n;r .We shall use the modulus of continuity Theorem 3.1.Suppose that f ∈ C p , p ∈ N 0 , is twice differentiable function on I and f and f belong to C p also.Then there exists M 7 (p) = const.> 0 such that for n ≥ 2p + 4.
Proof.For f satisfying our assumptions we have By elementary calculations we get and next by (1.1) and (1.12) we can write Using the inequality 2 for t, x > 0, and (2.1) we get and next by (2.3), (1.6), (1.13) and (1.7) we obtain (3.2) for n ≥ 2p + 4. Now we shall prove analogue of (1.4) for f ∈ C p .Theorem 3.2.Let p ∈ N 0 .Then there exists M 8 (p) = const.> 0 such that for every f ∈ C p and n ≥ 2p + 4 we have Similarly to [3] we apply the following Steklov function f h of f ∈ C p :
Similarly to [6] and [7] we use the following modified Taylor formula of f ∈ C r at a fixed point t > 0: where Hence we have which used to (3.10) implies that for x > 0 and n ≥ 2r + 2. Applying elementary properties of ω 1 defined by (3.1), we get from (3.11) From the above and by (1.6) and Lemma 2.1, we obtain n , for x > 0 and n ≥ 2r + 2, which by (1.7) gives (3.9).From Theorem 3.2 and Theorem 3.3 we can derive the following corollaries.
The above convergence is uniform on every interval [a, b], 0 < a < b < ∞.
Corollary 3.6.The order of approximation of f ∈ C r , r ≥ 2, by operators L n;r (f ) is better than by L n (f ).

3.2.
In this section we shall prove the Voronovskaya type theorem for operators L n;r , i.e. we shall give an analogue of (1.5).Theorem 3.7.Suppose that r ∈ N and f is a function belonging to C r and having derivatives f (r+1) and f (r+2) continuous and bounded on I. Then for every x > 0 we have as n → ∞.
Proof.Fix function f and x > 0. Then for every derivative f (j) , 0 ≤ j ≤ r, we can write the Taylor formula at x: where ϕ j (t) ≡ ϕ j (t, x) is function belonging to C 0 and lim t→x ϕ j (t) = ϕ j (x) = 0.
Hence F r defined by (1.10) can be written in the form By elementary calculations we get The properties of ϕ j , 0 ≤ j ≤ r, imply that Φ r given by (3.13) is a function belonging to C 0 and lim t→x Φ r (t) = Φ r (x) = 0. Hence there exists L n (Φ 2 r (t); x)
1) and (2.4) show that L n , n ≥ 2p, is a positive linear operator from the space C p into C p , p ∈ N 0 .Lemma 2.4.Let r ∈ N .Then there exists M 6