Cent. Eur. J. Math. • 8(1) • 2010 • 191-198 DOI: 10.2478/s11533-009-0057-9 Central European Journal of Mathematics On a q-analogue of Stancu operators Research Article Octavian Agratini1∗ 1 Faculty of Mathematics and Computer Science, Babeș-Bolyai University, Cluj-Napoca, Romania Received 8 June 2009; accepted 27 September 2009 Abstract: This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established. MSC: 41A36 Keywords: q-integers • q-Bernstein polynomials • Uniform convergence • Smoothness • Lipschitz-type maximal function © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction Recently, G. Nowak [13] introduced a q-analogue of Stancu’s operators [19]. As special case, the Phillips q-polynomials [16] can be reobtained. The aim of this note is to present approximation properties of the mentioned class of operators. The work is motivated by the following. During the last decade, q-Calculus was intensively used for the construction of various generalizations of many classical approximation processes of positive type. The first researches have been achieved by A. Lupaș [11] and G.M. Phillips [16] who proposed q-variants of the original Bernstein operators. Gaining popularity, these new polynomials have been studied lately by a number of authors, see, e.g., [6, 8, 15, 21, 22]. Also, other classes of discrete operators have been extended in q-Calculus, for example: Meyer-König and Zeller operators [20], Bleimann, Butzer and Hahn operators [5], Szász-Mirakjan operators [4], Balázs-Szabados operators [7]. 2. Background and preliminary results At first we collect some facts regarding q-Calculus, see, e.g., [3, 9]. Let q > 0. For any n ∈ N0 = {0} ∪ N, the q-integer [n]q and the q-factorial [n]q! are respectively defined by n−1 n X j Y [n]q = q ; [n]q! = [j]q; n ∈ N; j=0 j=1 ∗ E-mail: [email protected] 191 On a q-analogue of Stancu operators h n i and [0]q = 0, [0]q! = 1. The q-binomial coefficients are denoted by and are defined by k q h n i [n] ! = q ; k = 0; 1; : : : ; n: k q [k]q![n − k]q! h n i n Clearly, for q = 1 one has [n]1 = n, [n]1! = n! and = , the ordinary binomial coefficients. k q k In the sequel we always will assume that q ∈ (0; 1). For f ∈ C([0; 1]), α ≥ 0 and each n ∈ N, in [13] have been defined the operators n q;α X q;α [k]q (Bn f)(x) = pn;k (x)f ; x ∈ [0; 1]; (1) [n]q k=0 where Qk−1 Qn−1−k s q;α h n i i=0 (x + α[i]q) s=0 (1 − q x + α[s]q) pn;k (x) = n−1 : k q Q i=0 (1 + α[i]q) From here on, an empty product is taken to be equal to 1. This class contains as special cases the following well-known q;0 q sequences of linear and positive operators. For α = 0, Bn ≡ Bn represents q-Bernstein operator introduced by Phillips 1;0 1;α hαi [16]. For q = 1 and α = 0, Bn ≡ Bn is the classical Bernstein polynomial. For q = 1, Bn ≡ Bn turns into Stancu operator [19] defined as follows n 1 X n k (Bhαif)(x) = x[k;−α](1 − x)[n−k;−α]f ; x ∈ [0; 1]: (2) n 1[n;−α] k n k=0 m−1 Y Here t[m;a] = (t − ja) represents the generalized factorial power with the step a, a ∈ R, m ∈ N. The following j=0 identities hold [13] q;α q;α (Bn e0)(x) = 1; (Bn e1)(x) = x; q;α 1 x(1 − x) (Bn e2)(x) = x(x + α) + ; x ∈ [0; 1]; (3) 1 + α [n]q where ej , j ∈ N, stands for the monomial of j-th degree and e0(x) = 1. It is worth to be mentioned that other two q-analogues of Stancu operators have been earlier introduced by A. Lupaș [12]. Remark 2.1. (i) On the basis of (1) one ascertains that the approximated function f is evaluated at non-equally spaced points satisfying the relations k [k]q h n i k kq = ∈ ; n ; 0 ≤ k ≤ n: [n]q k qn n nq q;α (ii) These operators interpolate the approximated function f at the endpoints of the interval, this means (Bn f)(τ) = f(τ), τ ∈ {0; 1}: q;α q;α (iii) The operators are non-expansive. Indeed, since Bn e0 = 1, we get kBn fk ≤ kfk, where k · k represents the norm of the uniform convergence of the space C([0; 1]). hαi q q;α (iv) As the original Bernstein and Stancu operators verify Bnf ≤ Bn f, in the same manner one has Bnf ≤ Bn f, for every α ≥ 0 and f. 192 O. Agratini q;α The operator Bn can be reintroduced by using probabilistic tools. Following [2, Section 5.2], let (Ω; F;P) be a probability space and Zq : N × [0; 1] → M2(Ω) a random scheme on [0; 1], where M2(Ω) stands for the space of all real square-integrable random variables on Ω. We consider [k]q q;α P Zq(n; x) = = pn;k (x); 0 ≤ k ≤ n; n ∈ N; x ∈ [0; 1]: [n]q As usual, the mathematical expectation and the variance of Zq(n; x) are denoted by E(Zq(n; x)) and V ar(Zq(n; x)), respectively. Setting PZq(n;x) the distribution of Zq(n; x) with respect to P, one has Z Z 1 q;α (Bn f)(x) = f ◦ Zq(n; x)dP = fdPZq(n;x) = E(f(Zq(n; x))); (4) Ω 0 x ∈ [0; 1], and, in harmony with (3), we obtain 1 + α[n]q E(Zq(n; x)) = x; V ar(Zq(n; x)) = x(1 − x): (5) (1 + α)[n]q Theorem 2.1 ([13]). qn;αn Let (qn)n≥1, (αn)n≥1 be real sequences such that 0 < qn < 1, αn ≥ 0, n ∈ N. Let Bn , n ∈ N, be defined as in (1). If lim qn = 1 and lim αn = 0, then for each f ∈ C([0; 1]) n n qn;αn lim(Bn f)(x) = f(x); uniformly in x ∈ [0; 1]: (6) n The proof of this result, see [13, Theorem 2.6], is based on Bohman-Korovkin criterion. In Section 4 we will present another proof based on the probabilistic look over the operators. A similar approach has been achieved by Il’inskii and Ostrovska for q-Bernstein operators [8, Lemma 1]. 3. Statement of the results q;α We explore the rate of convergence of Bn , 0 < q < 1, α > 0, n ∈ N, in terms of the modulus of continuity ω1(f; ·), where ω1(f; δ) = sup |f(x) − f(y)|, δ ≥ 0. x;y∈[0;1] |x−y|≤δ Theorem 3.1. q;α Let Bn , n ∈ N, be defined by (1). (i) If f ∈ C([0; 1]), then s ! q;α 3 1 + α[n]q |(Bn f)(x) − f(x)| ≤ ω1 f; ; x ∈ [0; 1]: (7) 2 (1 + α)[n]q (ii) If f is differentiable and f0 ∈ C([0; 1]), then s s ! q;α 3 1 + α[n]q 0 1 + α[n]q |(Bn f)(x) − f(x)| ≤ ω1 f ; ; x ∈ [0; 1]: (8) 4 (1 + α)[n]q (1 + α)[n]q For q = 1, the inequalities (7), (8) reduce to the known estimates of the order of approximation by Stancu operators, see [19, Th. 5.1, Th. 5.2]. Taking q = qn ∈ (0; 1) and αn ≥ 0 such that lim qn = 1, lim αn = 0, we deduce lim[n]q = ∞ and relation (7) implies (6). n n n n 193 On a q-analogue of Stancu operators Remark 3.1. qn;αn p −1 Examining (7) one observes that the order of approximation of f by Bn is O( ([n]qn ) + αn depending on (qn)n − and (αn)n, two independent sequences. Since 0 < qn → 1 , there exist n0 ∈ N and a constant γ0 > 0 such that −1 n−1 −1 −1 ([n]qn ) ≤ (nqn ) < γ0n , for each n ≥ n0. To maximize 1/[n]qn it is necessary to have [n]qn ∼ n. This requirement is implied by the following: there exist n1 ∈ N and a constant γ1 > 0 such that qn > 1 − γ1/n for each n ≥ n1. Further −1 on, selecting√ (αn)n such that αn ∼ n , we deduce that, in any case, the order of approximation cannot be greater than O(1/ n), which represents exactly the order of approximation of f by the classical Bernstein operator Bn. q;α 1 q;α q;α m q;α q;α (m−1) q;α By definition, the m-th iterate of Bn is Bn := Bn , Bn := Bn ( Bn ), m = 2; 3;::: . Our next aim is to study m q;α the convergence of the iterates Bn as m tends to infinity. Theorem 3.2. q;α Let Bn , n ∈ N, be defined by (1). For any fixed n ∈ N, one has m q;α lim( Bn f)(x) = f(0) + (f(1) − f(0))x; f ∈ C([0; 1]); (9) m uniformly on [0; 1]. q We mention that for q-Bernstein polynomials Bn, n ∈ N, a detailed study of their iterates has been done by Ostrovska [14].