On a -Analogue of Stancu Operators

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].

Theorem 3.3. q;α q;α Let Bn , n ∈ N, be deﬁned by (1). If f is non-concave on [0; 1], then f ≤ Bn f for any n ∈ N.

q For q-Bernstein polynomials Bn, n ∈ N, the above result was established in [17, Th. 6]. µ λ As a consequence of this result, we deduce the following. Since the functions −e1, 0 < µ < 1, e1, λ > 1, are non-concave on [0; 1], for n ∈ N one has q;α µ µ q;α λ λ (Bn e1)(x) ≤ x ; (Bn e1)(x) ≥ x ; x ∈ [0; 1]:

q;α Clearly, we can consider Bn deﬁned on all measurable and bounded functions f on [0; 1]. We will characterize the local convergence for this positive linear operator by the elements of the Lipschitz class Lipβ. Here, the local behaviour of a function will be measured by the Lipschitz-type maximal function of order β introduced by B. Lenze [10] as

|f(x) − f(t)| ( ; ) = sup ∈ [0 1] ∈ (0 1] (10) ωeβ f x β ; x ; ; β ; : t∈[0;1] |x − t| t6=x

The ﬁniteness of ωeβ(f; ·) gives a local control for the smoothness of f. Boundedness of it is, roughly speaking, equivalent to f ∈ Lipβ on [0; 1]. We have the following local direct estimate.

Theorem 3.4. q;α Let β ∈ (0; 1] and f : [0; 1] → R be measurable and bounded. Let Bn , n ∈ N, be deﬁned by (1). For all x ∈ [0; 1] one has β q;α 2 2 |(Bn f)(x) − f(x)| ≤ (x − x ) hn(α; β)ωeβ(f; x);

β 1 1 2 where hn(α; β) = + . [n]q α

194 O. Agratini

4. Proofs

Proof of Theorem 2.1 (probabilistic approach). For proving the statement, it is enough to show the following. For any ε > 0, there exist ηε > 0 and nε ∈ N such that

qn;αn |(Bn f)(x) − f(x)| < ε; (11)

for all x ∈ [0; 1], qn ∈ (1 − ηε; 1), αn ∈ (0; ηε) and n ≥ nε. Let ε > 0 be arbitrarily ﬁxed. Since f ∈ C([0; 1]), a constant M exists such that |f(x)| ≤ M for all x ∈ [0; 1]. Let δ ∈ (0; 1] be chosen such that |f(t) − f(x)| < ε/2, whenever |t − x| < δ, t ∈ [0; 1], x ∈ [0; 1]. For all t and x belonging to [0; 1] such that |t − x| ≥ δ, we can write |f(t) − f(x)| ≤ 2M ≤ 2M(t − x)2/δ2. 2 Let ηε := εδ /M, where δ ≤ 1. The constant M can be taken as large as we want, consequently ηε < 1.

On the other hand, since lim αn = 0, lim qn = 1, we get lim[n]qn = ∞ and, for the above chosen ηε, nε ∈ N exists such n n n that 1 αn + < ηε; for each n ≥ nε: (12) [n]qn Taking in view both (4), (5) and (12) we can write successively

Z 1 qn;αn |(Bn f)(x) − f(x)| ≤ |f(t) − f(x)|PZqn (n;x)(dt) 0

Z Z ε Z 1 2M Z 1 = + ≤ + ( − )2 dPZqn (n;x) 2 t x dPZqn (n;x) 2 0 δ 0 t∈[0;1] t∈[0;1] |t−x|<δ |t−x|≥δ

ε 2M ε 2M 1 = + 2 V ar(Zqn (n; x)) ≤ + 2 αn + max x(1 − x) < ε: 2 δ 2 δ [n]qn x∈[0;1] Therefore, (11) is true.

Proof of Theorem 3.1. (i) Based on a result due to Shisha and Mond [18], for any linear positive operator

L : C([0; 1]) → B([0; 1]) the following inequality involving the modulus of continuity ω1(f; ·) takes place

|(Lf)(x) − f(x)| ≤ |f(x)||(Le0)(x) − 1|

1 p + (Le )(x) + (Lψ2)(x)(Le )(x) ω (f; δ); 0 δ x 0 1 x ∈ [0; 1], δ > 0, where ψx (t) = t − x, t ∈ [0; 1]. In view of (3), we obtain

q;α 2 1 + α[n]q (Bn ψx )(x) = x(1 − x): (13) (1 + α)[n]q

s 1 + α[n] Since x(1 − x) ≤ 1/4, x ∈ [0; 1], choosing δ = q , the conclusion follows. (1 + α)[n]q (ii) Under the assumption f0 ∈ C([0; 1]) one has [18]

0 |(Lf)(x) − f(x)| ≤ |f(x)||(Le0)(x) − 1| + |f (x)||(Lψx )(x)| p p 1 p + (Lψ2)(x) (Le )(x) + (Lψ2)(x) ω (f0; δ): x 0 δ x 1

q;α Since (Bn ψx )(x) = 0, with the same choosing of δ, the conclusion follows.

195 On a q-analogue of Stancu operators

Further on, we need the following result which was established by using the contraction principle.

Theorem 4.1 ([1]). Let Ln, n ∈ N, be deﬁned as follows

n X Ln : C([a; b]) → C([a; b]); (Lnf)(x) = ψn;k (x)f(xn;k ); k=0

where 0 = xn;0 < xn;1 < ··· < xn;n = b and for each 0 ≤ k ≤ n, ψn;k ∈ C([a; b]), ψn;k ≥ 0. We assume that Lnej = ej , j ∈ {0; 1}. Let us denote un := min (ψn;0(x) + ψn;n(x)): x∈[a;b] If un > 0, then m f(b) − f(a) lim( Lnf)(x) = f(a) + (x − a); f ∈ C([a; b]); m b − a uniformly on [a; b].

q;α Proof of Theorem 3.2. Choosing in Theorem 4.1 a = 0, b = 1, xn;k = [k]q/[n]q, ψn;k = pn;k , 0 ≤ k ≤ n, and q;α q;α knowing the identities (3), all is left to be proved is the relation min (pn;0 (x) + pn;n(x)) > 0. We get x∈[0;1]

n−1 !−1 n−1 n−1 ! q;α q;α Y Y Y s pn;0 (x) + pn;n(x)= (1 + α[i]) (x + α[s]) + (1 − q x + α[s]) i=0 s=0 s=0

n−1 !−1 n−1 !−1 Y 1 Y ≥ (1 + α[i]) (xn + (1 − x)n) ≥ (1 + α[i]) 2n−1 i=0 i=0

1 α −n ≥ 1 + ; 2n−1 1 − q consequently Theorem 4.1 can be applied and the proof of (9) is complete.

q;α Proof of Theorem 3.3. Let x ∈ [0; 1] arbitrarily be ﬁxed. We will prove f(x) ≤ (Bn f)(x). Since f is non-concave on [0; 1], on the basis of Jensen inequality, we can write

n ! n X X f λk tk ≤ λk f(tk ); (14) k=0 k=0

n X for any tk ∈ [0; 1] and λk ≥ 0, 0 ≤ k ≤ n, satisfying λk = 1. k=0 [k]q q;α Choosing tk = ∈ [0; 1] and λk = pn;k (x) ≥ 0, 0 ≤ k ≤ n, we deduce [n]q

n n X q;α X q;α λk = (Bn e0)(x) = 1 and λk tk = (Bn e1)(x) = x; k=0 k=0

see (3). Returning at (13), one obtains the desired result.

196 O. Agratini

Proof of Theorem 3.4. On the basis of (10), for all x ∈ [0; 1] and 0 ≤ k ≤ n, we have

β [k]q [k]q f(x) − f ≤ ωeβ(f; x) x − : [n]q [n]q

Consequently, we obtain

n [k] q;α X q;α q |f(x) − (Bn f)(x)| = pn;k (x) f(x) − f [n]q k=0 n β X q;α [k]q ≤ ωeβ(f; x) pn;k (x) x − : [n]q k=0

Applying Hölder’s inequality and taking into account (12), we get

q;α |f(x) − (Bn f)(x)| β 1− β n 2! 2 n ! 2 X q;α [k]q X q;α ≤ ωeβ(f; x) pn;k (x) x − pn;k (x) [n]q k=0 k=0 β q;α 2 2 = ωeβ(f; x)(Bn ψx ) (x) β 2 2 β 1 1 ≤ (x − x ) 2 + ωeβ(f; x); [n]q α and the proof is ﬁnished.

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