VORONOVSKAYA TYPE RESULTS AND OPERATORS FIXING TWO FUNCTIONS

The present paper deals with positive linear operators which fix two functions. The transfer of a given sequence (Ln) of positive linear operators to a new sequence (Kn) is investigated. A general procedure to construct sequences of positive linear operators fixing two functions which form an Extended Complete Chebyshev system is described. The Voronovskaya type formula corresponding to the new sequence which is strongly influenced by the nature of the fixed functions is obtained. In the last section our results are compared with other results existing in literature.

1 Introduction sequence (L n ) n≥1 satisfies a Voronovskaya type formula, i.e., lim n→∞ n (L n (f ; x) − f (x)) = a(x) where f ∈ D ∩ C 2 (I), x ∈ I. Such formulas are essential tools in approximation by positive linear operators. They are used to describe the rates of convergence and the saturation class. The iterates of certain positive linear operators can be used in order to approximate C 0 -semigroups of operators; in this case the Voronovskaya operator and the infinitesimal generator of the semigroup are strongly related (see [11,12]).
Usually one uses operators which preserve the constant functions; then c(x) = 0, x ∈ I. If, in addition, L n v = v, n ≥ 1, for a non-constant function v ∈ C 2 (I), then a(x) and b(x) are related by a(x)v (x) + b(x)v (x) = 0, x ∈ I.
In the last years, several papers were published, dealing with positive linear operators which fix two functions. A starting point was the paper [27] written by P.J. King who constructed positive linear operators on C[0, 1] fixing the constant function 1 and the function x 2 . Many generalizations followed, dealing with operators fixing 1 and a given function τ ; see Section 6. Recently, operators fixing two exponential functions were constructed: see [2,3,4,7,14,23].
The aim of our paper is twofold. On one hand, we investigate the transfer of a given sequence (L n ) of positive linear operators from an interval I to another interval J, and describe the Voronovskaya type formula corresponding to the new sequence (K n ). The properties of (K n ) inherited from (L n ), are also considered, as well as the inverse transfer from (K n ) to (L n ).
On the other hand, we propose a general procedure to construct sequences of positive linear operators fixing two functions which form an Extended Complete Chebyshev system. As mentioned above, the structure of the Voronovskaya operator is strongly influenced by the nature of the fixed functions. In a certain sense, the quality of the approximation offered by (L n ) can be expressed in terms of the corresponding Voronovskaya operator. We compare our results, from this point of view, with other results existing in the literature.
The basic definitions (in particular, the transfer of operators) are presented at the end of this section. Section 2 is devoted to the inverse transfer and the inherited properties. The Voronovskaya formula for the new operators is established in Section 3. Operators A n fixing two functions are constructed in Section 4. In Section 5, the convexity with respect to these two functions is characterized in terms of the operators A n and in terms of their Voronovskaya operator. Section 6 is devoted to examples and applications illustrating our general results and surveying some previous results existing in the literature.
We use notations usual in approximation theory by positive linear operators. As far as the domains of operators are concerned, we consider the maximal ones, i.e., we let the operators to act on the functions for which the involved series or integrals are convergent. We end this section with some basic definitions and notations.
Let I and J be intervals and D ⊆ C(I) a linear subspace containing the polynomial functions. By a slight abuse of notation we denote by e i , i = 0, 1, . . . , the function e i (t) = t i defined either on I or on J.
Let L n : D → C(I), n ≥ 1, be positive linear operators such that L n e 0 = e 0 , n ≥ 1. (1.1) Consider two continuous functions, u : Then v is injective, hence strictly monotone on J.
By using (1.1) and (1.2), we see that 2 Inverse transfer, iterates, commutativity iii) If L n L m = L m L n on H, then K n K m = K m K n on D 1 .
and this proves i).
ii) Let us prove that g • u ∈ H, for all g ∈ D 1 . (2.1) Indeed, if g ∈ D 1 , then g • u ∈ D. Moreover, u • v is the identity function on J, so that g • u = (g • u) • v • u, which proves (2.1). Since L n (H) ⊂ H, we have also from (2.1): For k = 1, ii) is obviously true. Suppose that ii) is true for a certain k. Then, for g ∈ D 1 , This induction argument proves ii). iii) Suppose that L n L m = L m L n on H. Let g ∈ D 1 . Then Using again (2.2) we see that L m (g • u) = (L m (g • u)) • v • u, which leads to

Transfer of Voronovskaya formula
In addition to the preceding hypotheses, in this section we suppose that u ∈ C 2 (I), v ∈ C 2 (J), and the sequence (L n ) satisfies the following Voronovskaya type formula: for all x ∈ I and f ∈ D ∩ C 2 (I), where α, β ∈ C(I) are two given functions.

Operators fixing two functions
for some strictly positive functions In this section we are concerned with positive linear operators fixing the two-dimensional linear subspace generated by an ECT-system. More precisely, consider the following setting. Let I be an interval, τ, γ ∈ C(I), γ : I → I bijective, τ (x) > 0, x ∈ I. Let L n : D → C(I), n ≥ 1, be positive linear operators, where D is a linear subspace of C(I) containing the polynomial functions. Suppose that L n e 0 = e 0 , L n e 1 = e 1 , n ≥ 1. (4.1) Then we have A n τ = τ, A n (γτ ) = γτ, n ≥ 1.
, and therefore {τ, τ (γ − γ(a))} is an ECT-system. The linear subspace generated by it is the same as the linear subspace generated by {τ, γτ }.
In order to present the Voronovskaya formula for the sequence (A n ) we need some additional hypotheses. In fact, we suppose that Moreover, assume that In particular, L n e 2 ≥ e 2 , and (4.6) leads to Theorem 3. Under the above assumptions, if f ∈ D 2 ∩ C 2 (I) and x ∈ I, then Due to (4.6), we can apply Theorem 2 with β = 0. It follows that and the proof is finished.
On the other hand, consider the differential operator .
It is easy to verify that Consequently, we have Moreover, Theorem 2 shows that if f ∈ D 2 ∩ C 2 (I) and x ∈ I, then According to (4.7), α(x) ≥ 0, x ∈ I. Suppose that this inequality is strict at each interior point of I. Now we can prove Since γ is strictly increasing on I, we have ϕ(t, x) > 0 if t > x, and ϕ(t, x) < 0 if t < x.
Let a ∈ I. The above inequalities show that t a ϕ(t, x)Af (x)dx ≥ 0, t ∈ I.
On the other hand, Combined with (5.2), this yields

Now using (4.3), we get
In particular, A n f (a) ≥ f (a), n ≥ 1, for each a ∈ I. This means that A n f ≥ f , and the proof is complete.
Remark 3. i) Let p be a polynomial function. Then and in many cases we can compute L n (p; ·). ii) Denote ϕ n := L n e 2 − e 2 . Then |L n (f ; and consequently A n (f ; x) ≥ f (x), x ∈ I.

Examples and applications
In this section we present some examples illustrating the preceding general results.
be the Meyer-König and Zeller operators (see [11,28]). It is well known that L n e 0 = e 0 and L n e 1 = e 1 . In this context the operators K n introduced by (1.3) are described as dx 2 (see [33]). According to Theorem 3.1, the Voronovskaya operator for (K n ) is t(1+t) 2 d 2 dt 2 + t d dt . As mentioned above, condition (4.1) is satisfied. Choosing in Section 4 τ = e 0 and γ = v, we have γ −1 = u. Consequently, the operators K n can be described as in (4.2), and we see that they fix the functions e 0 and v (see (4.3)).
Remark 4. The operators K n given by (6.1) should be compared with the classical Baskakov operators These operators fix the functions e 0 and e 1 . A way to represent the MKZ operators in terms of Baskakov operators can be found in [9]. The MKZ operators L n can be represented in terms of the operators K n defined by (6.1) if we use Theorem 1 (i). Starting with the Baskakov operators H n , let us construct the operators K n (f ; x) := H n (f • v; u(x)). Theñ They fix the functions e 0 and u. With I, J, u, v, D 1 as in Example 1, the corresponding operators K n will be K n (f ; t) = n k=0 n k t k (1 + t) n g k n − k , g ∈ D 1 , t ≥ 0. (6.5) In the above sum, the last value of g is g(∞).

The Voronovskaya operator for the Bernstein operators is
Theorem 2 shows that the corresponding Voronovskaya operator for the sequence The operators K n fix the functions e 0 and v. The asymptotic behavior of the iterates of L n from (6.4), respectively K n from (6.5) is the same as in (6.2), (6.3).
Remark 5. The operators K n from (6.5) should be compared with the classical Bleimann-Butzer-Hahn operators A representation of the BBH operators in terms of the Bernstein operators can be found in [9]. Starting with the BBH operators G n , let us construct the operatorsK n (f ; They fix e 0 and u. Example 3. Consider the Baskakov-Durrmeyer operators It is known that L n L m = L m L n , m, n ≥ 1 (see [24, p.14]). Since (v • u)(x) = x, we can apply Theorem 1 (iii) to conclude that K n K m = K m K n , m, n ≥ 1.
Example 4. Let L n : C 2π [−π, π] → C 2π [−π, π] be the De la Vallée Poussin operators, i.e., It is elementary to prove that and L n (H) ⊂ H. Moreover, L n L m = L m L n on H. Therefore, all the conclusions i), ii), iii) of Theorem 1 are valid in this context. For details and several applications, see [8,Section 5].
are the Weierstrass operators (see [11]). The domain D of L n contains, in particular, the bounded continuous functions and the polynomial functions defined on R.
It is elementary to prove that Remark 6. Consider two sequences of positive linear operators (U n,j ) n≥1 , j = 1, 2, on the same interval I. Suppose that they fix the same two functions w 1 , w 2 : U n,j w i = w i , n ≥ 1, i, j = 1, 2. (6.6) Moreover, suppose that each sequence satisfies a Voronovskaya type formula. Due to (6.6), these formulas are of the form where the functions p(x) and q(x) satisfy the system Thus we have So, in this sense, if on a certain subinterval I 0 we have |ω 1 (x)| ≤ |ω 2 (x)|, x ∈ I 0 , then the quality of the approximation on I 0 offered by (U n,1 ) n≥1 is better than offered by (U n,2 ) n≥1 .
Example 9. Another generalization of Example 7 can be found in [10]. The authors of that article constructed a sequence of positive linear operators on C[0, 1] fixing e 0 and e j , for a fixed integer j ≥ 1. It was conjectured in [18] that the corresponding Voronovskaya operator is This conjecture was validated by M. Birou [15].
Example 10. Another sequence of positive linear operators fixing e 0 and e j was introduced in [16]. It has the Voronovskaya operator were considered in [17].
They fix e 0 and τ . It was proved there that the associated Voronovskaya operator is Clearly, the sequence (B τ n ) n≥1 is a particular case of the sequence (K n ) n≥1 from (1.3). If we take τ = e j on (0, 1], we get the operator which should be compared in the sense of Remark 6 with the operator from Example 9, i.e., More precisely, we have to solve the inequation , with x ∈ (0, 1]. (6.8) It is easy to see that there exists x j ∈ (0, 1) such that (6.8) is satisfied if and only if x ∈ [x j , 1].
Example 13. The operators B n f ( √ t); x 2 were used in [17] in order to solve a problem raised in [21]. They fix e 0 and e 2 , and have the Voronovskaya operator Example 14. If in Section 4 we take γ = τ , we get operators fixing τ and τ 2 . Such operators were constructed also in [4].