Statistical approximation properties of λ-Bernstein operators based on q-integers

Qing-Bo Cai; Guorong Zhou; Junjie Li

doi:10.1515/math-2019-0039

Article Open Access

Statistical approximation properties of λ-Bernstein operators based on q-integers

Qing-Bo Cai , Guorong Zhou and Junjie Li

Published/Copyright: May 30, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we introduce a new generalization of λ-Bernstein operators based on q-integers, we obtain the moments and central moments of these operators, establish a statistical approximation theorem and give an example to show the convergence of these operators to f(x). It can be seen that in some cases the absolute error bounds are smaller than the case of classical q-Bernstein operators to f(x).

Keywords: q-integers; λ-Bernstein operators; Statistical convergence; basis function

MSC 2010: 41A10; 41A25; 41A36

1 Introduction

A generalization of Bernstein polynomials based on q-integers was proposed by Lupaş in 1987 in [1]. However, the Lupaş q-Bernstein operators are rational functions rather than polynomials. The Phillips q-Bernstein polynomials were introduced by Phillips in 1997 in [2]. After that, there are many papers mentioning the approximation properties of positive linear operators in the area of q-calculus due to its applications in the field of approximation theory, such as generalized q-Bernstein polynomials [3, 4, 5, 6], Durrmeyer type q-Bernstein operators [7, 8, 9], Kantorovich type q-Bernstein operators [10, 11, 12] and so on.

As we know, Phillips [2] defined the following q-Bernstein operators:

Bn(f;x)=∑i=0nbn,i(x;q)f[i]q[n]q, (1)

where b_n,i(x; q) (i = 0, 1, …, n) are q-Bernstein basis functions and defined as

bn,i(x;q)=niqxi∏s=0n−i−11−qsx. (2)

Motivated by above work, in this paper, we introduce λ-Bernstein operators based on q-integers as

B~n,q,λ(f;x)=∑i=0nb~n,i(x;q,λ)f[i]q[n]q, (3)

where

b~n,0(x;q,λ)=bn,0(x;q)−λ[n]q+1bn+1,1(x;q),b~n,i(x;q,λ)=bn,i(x;q)+λ[n]q−2[i]q+1[n]q2−1bn+1,i(x;q)−[n]q−2q[i]q−1[n]q2−1bn+1,i+1(x;q),(i=1,2,...,n−1)b~n,n(x;q,λ)=bn,n(x;q)−λ[n]q+1bn+1,n(x;q), (4)

b_n,i(x; q) are defined in (2), λ ∈ [–1, 1], n ≥ 2, x ∈ [0, 1] and 0 < q ≤ 1.

Obviously, when λ = 0, B͠_n,q,0(f; x) turn out to be q-Bernstein operators defined in (1); when λ = 0, q = 1, B͠_n,1,0(f; x) turn out to be classical Bernstein operators; when q = 1, B͠_n,1,λ(f; x) turn out to be the λ-Bernstein operators which are studied by Cai, et al. in [13].

Details of q-integers can be found in [14, 15]. For any fixed real number 0 < q ≤ 1 and each nonnegative integer k, we denote q-integers by [k]_q, where [k]q:=1−qk1−q,q≠1,k,q=1. Also q-factorial and q-binomial coefficients are defined as follows:

[k]q!:=[k]q[k−1]q...[1]q,k=1,2,...,1,k=0,,nkq:=[n]q![k]q![n−k]q!(n≥k≥0).

Now we recall the concepts of statistical convergence.

Let K be a subset of ℕ, the set of all natural numbers. The density of K is defined by

δ(K):=limn→∞1n∑k=1nχK(k),

provided the limit exists [16], where χ_K is the characteristic function of K.

A sequence x : = {x_k} is called statistically convergent to a number L, if for every ϵ > 0, δ (k ∈ ℕ: |x_k – L| ≥ ϵ) = 0 [17]. This convergence is denoted as st – lim_n→∞ x_n = L.

We also recall the following theorem given by Gadjiev and Orhan [18].

Theorem 1.1

(See [18]) If the sequence of positive linear operators A_n : C_M[a, b] → C[a, b] satisfies the conditions st – lim_n→∞||A_n(e_v;⋅) – e_v||_{C[a, b]} = 0, with e_v(t) = t^v for v = 0, 1, 2, then for any function f ∈ C_M[a, b], we have

st−limn→∞||An(f;⋅)−f||C[a,b]=0,

where C_M[a, b] denotes the space of all functions f which are continuous in [a, b] and bounded on the all positive axis.

The aims of this paper are to introduce a new generalization of λ-Bernstein operators based on q-integers and give the statistical approximation properties of these operators using the concept of statistical convergence.

The rest of this paper is organized as follows. In section 2, we give some lemmas which are needed to prove our main results. In section 3, we obtain the statistical approximation theorem and give an example to show that, in some cases, the absolute error bounds of these operators defined in (3) to f(x) are smaller than the bounds of classical q-Bernstein operators to f(x).

2 Some lemmas

In the sequel, let e_v(t) = t^v for v = 0, 1, 2. In order to obtain the main results, we need the following lemmas:

Lemma 2.1

Let λ ∈ [–1, 1], x ∈ [0, 1] and q ∈ (0, 1], for the operators B͠_n,q,λ(f; x), we have

B~n,q,λ(e0;x)=1. (5)

Proof

Using [n]_q = 1 + q[n – 1]_q, we have

∑i=0nb~n,i(x;q,λ)=∑i=0nbn,i(x;q)−λ[n]q+1bn+1,1(x;q)+λ[n]q−2[1]q+1[n]q2−1bn+1,1(x;q) −λ[n]q−2q[1]q−1[n]q2−1bn+1,2(x;q)+λ[n]q−2[2]q+1[n]q2−1bn+1,2(x;q) −λ[n]q−2q[2]q−1[n]q2−1bn+1,3(x;q)+...+λ[n]q−2[n−1]q+1[n]q2−1bn+1,n−1(x;q) −λ[n]q−2q[n−1]q−1[n]q2−1bn+1,n(x;q)−λ[n]q+1bn+1,n(x;q) =∑i=0nbn,i(x;q)+2q[n−1]q+1−[n]q[n]q2−1λbn+1,n(x;q)−λ[n]q+1bn+1,n(x;q) =∑i=0nbn,i(x;q)=1.

Then we can obtain (5) since B͠_n,q,λ(e₀; x) = ∑k=0n b͠_n,k(x; q, λ).□

Lemma 2.2

Let λ ∈ [–1, 1], x ∈ [0, 1] and q ∈ (0, 1], for the operators B͠_n,q,λ(f; x), we have

B~n,q,λ(e1;x)=x+[n+1]qx1−xnλ[n]q[n]q−1−2[n+1]qxλ[n]q2−11−xn[n]q+qx1−xn−1 +λq[n]q[n]q+11−∏s=0n1−qsx−xn+1−[n+1]qx1−xn +λ[n]q2−12[n+1]qx21−xn−1−2[n+1]qxq[n]q1−xn +2q[n]q1−∏s=0n1−qsx−xn+1. (6)

Proof

From (3), we have

B~n,q,λ(e1;x)=∑i=0nb~n,i(x;q,λ)[i]q[n]q=∑i=1n−1[i]q[n]qb~n,i(x;q,λ)+b~n,n(x;q,λ)=∑i=1n−1[i]q[n]qbn,i(x;q)+λ[n]q−2[i]q+1[n]q2−1bn+1,i(x;q)−[n]q−2q[i]q−1[n]q2−1bn+1,i+1(x;q)+bn,n(x;q)−λ[n]q+1bn+1,n(x;q)=∑i=0n[i]q[n]qbn,i(x;q)+λ∑i=0n[i]q[n]q[n]q−2[i]q+1[n]q2−1bn+1,i(x;q)−λ∑i=1n−1[i]q[n]q[n]q−2q[i]q−1[n]q2−1bn+1,i+1(x;q).

As we know, for the q-Bernstein operators [2], we have ∑i=0n[i]q[n]qbn,i(x;q)=x, then

B~n,q,λ(e1;x)=x+[n+1]qλ[n]q[n]q−1∑i=1n[i]q[n+1]q[n+1]q![i]q![n+1−i]q!xi∏s=0n−i1−qsx −2[n+1]qλ[n]q2−1∑i=1n[i]q2[n]q[n+1]q[n+1]q![i]q![n+1−i]q!xi∏s=0n−i1−qsx −[n+1]qλ[n]q[n]q+1∑i=1n−1[i]q[n+1]q[n+1]q![i+1]q![n−i]q!xi+1∏s=0n−i−11−qsx +2q[n+1]qλ[n]q2−1∑i=1n−1[i]q2[n]q[n+1]q[n+1]q![i+1]q![n−i]q!xi+1∏s=0n−i−11−qsx:=x+△1+△2+△3+△4. (7)

Firstly, we have

△1=[n+1]qλ[n]q[n]q−1∑i=1n[i]q[n+1]q[n+1]q![i]q![n+1−i]q!xi∏s=0n−i1−qsx =[n+1]qλx[n]q[n]q−1∑i=0n−1bn,i(x;q)=[n+1]qx1−xnλ[n]q[n]q−1. (8)

Secondly, since [i]q2=[i]q+q[i]q[i−1]q, we get

△2=−2[n+1]qλ[n]q2−1∑i=1n[i]q2[n]q[n+1]q[n+1]q![i]q![n+1−i]q!xi∏s=0n−i1−qsx =−2[n+1]qλ[n]q[n]q2−1∑i=1n[i]q[n+1]q[n+1]q![i]q![n+1−i]q!xi∏s=0n−i1−qsx +−2q[n+1]qλ[n]q2−1∑i=1n[i]q[i−1]q[n+1]q[n]q[n+1]q![i]q![n+1−i]q!xi∏s=0n−i1−qsx =−2[n+1]qλx[n]q[n]q2−1∑i=0n−1bn,i(x;q)+−2q[n+1]qx2λ[n]q2−1∑i=0n−2bn−1,i(x;q) =−2[n+1]qxλ[n]q2−11−xn[n]q+qx1−xn−1. (9)

Thirdly, since [i]_q = [i + 1]_q/q – 1/q, we have

△3=−[n+1]qλ[n]q[n]q+1∑i=1n−1[i]q[n+1]q[n+1]q![i+1]q![n−i]q!xi+1∏s=0n−i−11−qsx =−[n+1]qλq[n]q[n]q+1∑i=1n−1[i+1]q[n+1]q[n+1]q![i+1]q![n−i]q!xi+1∏s=0n−i−11−qsx +λq[n]q+1[n]q∑i=1n−1bn+1,i+1(x;q) =−[n+1]qxλq[n]q[n]q+1∑i=1n−1bn,i(x;q)+λq[n]q+1[n]q∑i=1n−1bn+1,i+1(x;q) =−[n+1]qxλq[n]q[n]q+11−∏s=0n−11−qsx−xn +λq[n]q[n]q+11−∏s=0n1−qsx−[n+1]x∏s=0n−11−qsx−xn+1 =λq[n]q[n]q+11−∏s=0n1−qsx−xn+1−[n+1]qx1−xn. (10)

Finally, since [i]q2 = [i]_q[i + 1]_q/q – [i]_q/q, we have

△4=2q[n+1]qλ[n]q2−1∑i=1n−1[i]q2[n]q[n+1]q[n+1]q![i+1]q![n−i]q!xi+1∏s=0n−i−11−qsx=2[n+1]qλ[n]2−1∑i=0n−2bn−1,i(x;q)−2[n+1]qxλq[n]q[n]q2−1∑i=1n−1bn,i(x;q)+2λq[n]q[n]q2−1∑i=1n−1bn+1,i+1(x;q)=λ[n]q2−12[n+1]qx21−xn−1−2[n+1]qxq[n]q1−xn+2q[n]q1−∏s=0n1−qsx−xn+1. (11)

Then, (6) can be obtained by (7)-(11).□

Lemma 2.3

Let λ ∈ [–1, 1], x ∈ [0, 1], n > 1 and q ∈ (0, 1], for the operators B͠_n,q,λ(f; x), we have

B~n,q,λ(e2;x)=x2+x(1−x)[n]q+[n+1]qxλ[n]q[n]q−1qx1−xn−1+1−xn[n]q−2[n+1]qλ[n]q[n]q2−1×x1−xn[n]q+q(2+q)x21−xn−1+q3[n−1]qx31−xn−2−λq[n]q[n]q+1[n+1]qx21−xn−1−[n+1]qx1−xnq[n]q+1−∏s=0n1−qsx−xn+1q[n]q+2λ[n]q[n]q2−1×q[n−1]q[n+1]qx31−xn−2−(1−q)[n+1]qx21−xn−1q+[n+1]qx1−xnq2[n]q−1−∏s=0n1−qsx−xn+1q2[n]q. (12)

Proof

From (3), we have

B~n,q,λ(e2;x)=∑i=0n[i]q2[n]q2b~n,i(x;q,λ)=∑i=0n[i]q2[n]q2bn,i(x;q)+λ∑i=0n[i]q2[n]q2[n]q−2[i]q+1[n]q2−1bn+1,i(x;q)−λ∑i=1n−1[i]q2[n]q2[n]q−2q[i]q−1[n]q2−1bn+1,i+1(x;q).

For the q-Bernstein polynomials [2], we have ∑i=0n[i]q2[n]q2bn,i(x;q)=x2+x(1−x)[n]q, thus

B~n,q,λ(e2;x)=x2+x(1−x)[n]q+λ[n]q−1∑i=0n[i]q2[n]q2bn+1,i(x;q)−2λ[n]q2−1∑i=0n[i]q3[n]q2bn+1,i(x;q)−λ[n]q+1∑i=1n−1[i]q2[n]q2bn+1,i+1(x;q)+2qλ[n]q2−1∑i=1n−1[i]q3[n]q2bn+1,i+1(x;q):=x2+x(1−x)[n]q+△5+△6+△7+△8. (13)

We have

△5=λ[n]q−1∑i=0n[i]q2[n]q2bn+1,i(x;q) =[n+1]qqx2λ[n]q[n]q−1∑i=0n−2bn−1,i(x;q)+[n+1]qxλ[n]q2[n]q−1∑i=0n−1bn,i(x;q) =[n+1]qxλ[n]q[n]q−1qx1−xn−1+1−xn[n]q. (14)

Since [i]q3 = [i]_q + q(2 + q)[i]_q[i – 1]_q + q³[i]_q[i – 1]_q[i – 2]_q, we have

△6=−2λ[n]q2−1∑i=0n[i]q3[n]q2bn+1,i(x;q) =−2[n+1]qxλ[n]q2[n]q2−1∑i=0n−1bn,i(x;q)−2q(2+q)[n+1]qx2λ[n]q[n]q2−1∑i=0n−2bn−1,i(x;q) −2q3[n−1]q[n+1]qx3λ[n]q[n]q2−1∑i=0n−3bn−2,i(x;q) =−2[n+1]qλ[n]q[n]q2−1x1−xn[n]q+q(2+q)x21−xn−1+q3[n−1]qx31−xn−2. (15)

Next, since [i]q2 = [i + 1]_q[i]_q/q – [i + 1]_q/q² + 1/q², we get

△7=−λ[n]q+1∑i=1n−1[i]q2[n]q2bn+1,i+1(x;q) =−[n+1]qx2λq[n]q[n]q+1∑i=0n−2bn−1,i(x;q)+[n+1]qxλq2[n]q2[n]q+1∑i=1n−1bn,i(x;q) −λq2[n]q2[n]q+1∑i=1n−1bn+1,i+1(x;q) =−λq[n]q[n]q+1[n+1]qx21−xn−1−[n+1]qx1−xnq[n]q +1−∏s=0n1−qsx−xn+1q[n]q. (16)

Finally, by some computations, we have

[i]q3=[i+1]q[i]q[i−1]q−1−qq2[i+1]q[i]q+[i+1]qq3−1q3.

Then, we obtain

△8=2qλ[n]q2−1∑i=1n−1[i]q3[n]q2bn+1,i+1(x;q) =2q[n−1]q[n+1]qx3λ[n]q[n]q2−1∑i=0n−3bn−2,i(x;q)−2(1−q)[n+1]qx2λq[n]q[n]q2−1∑i=0n−2bn−1,i(x;q) +2[n+1]qxλq2[n]q2[n]q2−1∑i=1n−1bn,i(x;q)−2λq2[n]q2[n]q2−1∑i=1n−1bn+1,i+1(x;q) =2λ[n]q[n]q2−1q[n−1]q[n+1]qx31−xn−2−(1−q)[n+1]qx21−xn−1q +[n+1]qx1−xnq2[n]q−1−∏s=0n1−qsx−xn+1q2[n]q. (17)

Using (13)-(17), we can obtain (12), Lemma 2.1 is proved.□

Remark 2.4

When λ = 0, we get the moments of q-Bernstein operators (see [2]).

Lemma 2.5

For λ ∈ [–1, 1], x ∈ [0, 1] and q ∈ (0, 1], the operators B͠_n,q,λ(f; x) are positive linear operators.

Proof

Indeed, we only need to prove b͠_n,i(t; q, λ) ≥ 0 for 0 ≤ i ≤ n. For i = 0, we have

b~n,0(x;q,λ)=bn,0(x;q)−λ[n]q+1bn+1,1(x;q)=∏s=0n−11−qsx−λ[n]q+1[n+1]qx∏s=0n−11−qsx=∏s=0n−11−qsx1−[n+1]q[n]q+1λx≥0,

since [n]_q + 1 ≥ [n + 1]_q. Similarly, we can obtain b͠_n,n(x; q, λ) ≥ 0. Then we will prove b͠_n,i(x; q, λ)≥ 0, where 1 ≤ i ≤ n – 1. Actually, by (4), we have

b~n,i(x;q,λ)=bn,i(x;q)+λ[n]q−2[i]q+1[n]q2−1bn+1,i(x;q)−[n]q−2q[i]q−1[n]q2−1bn+1,i+1(x;q)=bn,i(x;q)1+λ[n]q−2[i]q+1[n]q2−1[n+1]q[n+1−i]q1−qn−ix−[n]q−2q[i]q−1[n]q2−1[n+1]q[i+1]qx.

We need to prove

−1≤[n]q−2[i]q+1[n]q2−1[n+1]q[n+1−i]q1−qn−ix−[n]q−2q[i]q−1[n]q2−1[n+1]q[i+1]qx≤1. (18)

We discuss the above inequality in three cases as follows:

For 1 ≤ [i]_q ≤ min [n]q−12q,[n]q+12, we have

0≤[n+1]q[n+1−i]q[n]q−2[i]q+1[n]q2−1,[n+1]q[i+1]q[n]q−2q[i]q−1[n]q2−1≤1.

Then we can obtain (18) by the fact that 0 ≤ 1 – q^n–ix, 0 ≤ x ≤ 1.
For max [n]q+12,[n]q−12q≤[i]q≤[n−1]q, we have

−1≤[n+1]q[n+1−i]q[n]q−2[i]q+1[n]q2−1,[n+1]q[i+1]q[n]q−2q[i]q−1[n]q2−1≤0.

Then we can get (18) for 0 ≤ 1 – q^n–ix, 0 ≤ x ≤ 1.
For min[n]q+12,[n]q−12q<[i]q<max[n]q+12,[n]q−12q.

If [n]q−12q<[i]q<[n]q+12, we have

0<[n]q−2[i]q+1[n]q2−1[n+1]q[n+1−i]q, −[n]q−2q[i]q−1[n]q2−1[n+1]q[i+1]q≤12;

If [n]q+12<[i]q<[n]q−12q, we have

−12≤[n]q−2[i]q+1[n]q2−1[n+1]q[n+1−i]q,−[n]q−2q[i]q−1[n]q2−1[n+1]q[i+1]q<0.

Then we can get(18) in both two subcases. Therefore, we get the proof of Lemma 2.5.□

3 Main results

Let us give the following statistical approximation theorem for the operators (3).

Theorem 3.1

Let q = {q_n} be a sequence satisfying st – lim_n→∞ q_n = 1 for 0 < q_n < 1. Then, for all f ∈ C[0, 1], λ ∈ [–1, 1], n > 1 and the operators B͠_{n,q_n,λ}(f; x), we have

st−limn→∞B~n,qn,λ(f;⋅)−f=0. (19)

Proof

By Lemma 2.1, it is clear that

st−limn→∞B~n,qn,λ(e0;⋅)−e0C[0,1]=0. (20)

In order to obtain the desired result, using Lemma 2.5 and Theorem 1.1, we only need to prove the following equalities

st−limn→∞B~n,qn,λ(ev;⋅)−evC[0,1]=0, v=1,2. (21)

For v = 1, using Lemma 2.2, we have

B~n,qn,λ(e1;x)−e1(x)≤[n+1]qn[n]qn[n]qn−1+2[n+1]qn[n]qn2−11[n]qn+1+1qn[n]qn[n]qn+1+1+[n+1]qnqn[n]qn[n]qn+1+2[n]qn2−1[n+1]qn+[n+1]qnqn[n]qn+1qn[n]qn≤qn[n+1]qn+2qn+4qn[n]qn[n]qn−1+5[n]qn−1+2qn[n]qn[n]qn2−1,

since [n + 1]_{q_n} ≤ [n]_{q_n} + 1, we obtain

B~n,qn,λ(e1;x)−e1(x)≤6[n]qn−1+3qn+4qn[n]qn[n]qn−1+2qn[n]qn[n]qn2−1.

For a given ϵ > 0, let us define the following sets

U=k:B~k,qk,λ(e1;⋅)−e1C[0,1]≥ϵ; U1=k:6[k]qk−1≥ϵ3;U2=k:3qk+4qk[k]qk[k]qk−1≥ϵ3; U3=k:2qk[k]qk[k]qk2−1≥ϵ3.

It is obvious that U ⊂ U₁ ∪ U₂ ∪ U₃, which implies that

δk≤n:B~k,qk,λ(e1;⋅)−e1C[0,1]≥ϵ≤δk≤n:6[k]qk−1≥ϵ3+δk≤n:3qk+4qk[k]q[k]qk−1≥ϵ3+δk≤n:2qk[k]qk[k]qk2−1≥ϵ3. (22)

Since st – lim_n→∞ q_n = 1, we have

st−limn→∞6[n]qn−1=0; st−limn→∞3qn+4qn[n]qn[n]qn−1=0;st−limn→∞2qn[n]qn[n]q2−1=0,

that is to say, the right hand side of (22) is zero. Hence, (21) is proved for v = 1.

For v = 2, by Lemma 2.3, we have

B~n,qn,λ(e2;x)−e2(x)≤14[n]qn+[n+1]qn[n]qn[n]qn−11+1[n]qn+2[n+1]qn[n]qn[n]qn2−11[n]qn+3+[n−1]qn+1qn[n]qn[n]qn+1[n+1]qn+[n+1]qnqn[n]qn+1qn[n]qn+2[n]qn[n]qn2−1[n−1]qn[n+1]qn+[n+1]qnqn+[n+1]qnqn2[n]qn+1qn2[n]qn≤14[n]qn+5[n]qn−1+1qn[n]qn+2[n]qn[n]qn−1+8[n]qn2−1+1qn2[n]qn2+2qn[n]qn[n]qn−1+1[n]qn2[n]qn−1+8[n]qn[n]qn2−1+1qn2[n]qn2[n]qn+1+2qn2[n]qn2[n]qn−1+2[n]qn2[n]qn2−1+2qn2[n]qn2[n]qn2−1.

For a given ϵ > 0, we define the following sets

V=k:B~k,qk,λ(e2;⋅)−e2C[0,1]≥ϵ;V1=k:14[k]qk+5[k]qk−1+1qk[k]qk≥ϵ4;V2=k:2[k]qk[k]qk−1+8[k]qk2−1+1qk2[k]qk2+2qk[k]qk[k]qk−1≥ϵ4;V3=k:1[k]qk2[k]qk−1+8[k]qk[k]qk2−1+1qk2[k]qk2[k]qk+1+2qk2[k]qk2[k]qk−1≥ϵ4;V4=k:2[k]qk2[k]qk2−1+2qk2[k]qk2[k]qk2−1≥ϵ4.

Obviously, V ⊂ V₁ ∪ V₂ ∪ V₃ ∪ V₄, then one have

δk≤n:B~k,qk,λ(e2;⋅)−e2C[0,1]≥ϵ≤δk≤n:14[k]qk+5[k]qk−1+1qk[k]qk≥ϵ4+δk≤n:2[k]qk[k]qk−1+8[k]qk2−1+1qk2[k]qk2+2qk[k]qk[k]qk−1≥ϵ4+δk≤n:1[k]qk2[k]qk−1+8[k]qk[k]qk2−1+1qk2[k]qk2[k]qk+1+2qk2[k]qk2[k]qk−1≥ϵ4+δk≤n:2[k]qk2[k]qk2−1+2qk2[k]qk2[k]qk2−1≥ϵ4. (23)

Since st – lim_n→∞ q_n = 1, we obtain the following equalities

st−limn→∞14[n]qn+5[n]qn−1+1qn[n]qn=0;st−limn→∞2[n]qn[n]qn−1+8[n]qn2−1+1qn2[n]qn2+2qn[n]qn[n]qn−1=0;st−limn→∞1[n]qn2[n]qn−1+8[n]qn[n]qn2−1+1qn2[n]qn2[n]qn+1+2qn2[n]qn2[n]qn−1=0;st−limn→∞2[n]qn2[n]qn2−1+2qn2[n]qn2[n]qn2−1=0,

that is, the right hand side of (23) is zero. Therefore, (21) is proved for v = 2. Theorem 3.1 is proved.□

Finally, we give an example to show the convergence of B͠_n,q,λ(f; x) to f(x) with different values of parameters, we also obtain the absolute error bounds in some cases.

Example 3.2

Let f(x) = 1 – cos(4e^x), the graphs of B͠_n,q,–1(f; x) and B͠_n,q,1(f; x) with different cases of n and q are shown in Figure 1. Figure 2 shows the graphs of B͠_n,q,1(f; x) and B͠_n,q,0(f; x) with n = 10 and q = 0.9. In Table 1, we give the absolute error bounds of B͠_n,q,λ(f; x) to f(x).

$Figure 1 Convergence of B͠n,q,λ(f; x) for λ = –1, λ = 1 and different values of q and n.$

Figure 1

Convergence of B͠_n,q,λ(f; x) for λ = –1, λ = 1 and different values of q and n.

$Figure 2 Convergence of B͠10,0.9,λ(f; x) for λ = 1 and λ = 0.$

Figure 2

Convergence of B͠_10,0.9,λ(f; x) for λ = 1 and λ = 0.

Table 1

The errors of the approximation of B͠_n,q,λ(f; x) to f(x).

			∥f – B͠_n,q,λ(f)∥_∞
		n = 10	n = 20	n = 50	n = 100
λ = –1	q = 0.9	0.586703	0.469244	0.425725	0.423995
	q = 0.99	0.447241	0.259814	0.128908	0.082675
	q = 0.999	0.438761	0.244595	0.107170	0.056663
	q = 0.9999	0.437909	0.243088	0.105111	0.054359

λ = 0	q = 0.9	0.562861	0.459109	0.418998	0.417386
	q = 0.99	0.432402	0.256685	0.128892	0.082737
	q = 0.999	0.423813	0.241516	0.107168	0.056722
	q = 0.9999	0.422953	0.240017	0.105112	0.054418

λ = 1	q = 0.9	0.539161	0.449016	0.412299	0.410807
	q = 0.99	0.418391	0.253900	0.128901	0.082802
	q = 0.999	0.409770	0.238817	0.107201	0.056785
	q = 0.9999	0.408905	0.237324	0.105145	0.054481

One can see from Table 1 that in some cases, such as n = 10, 20 and λ = 1, the absolute error bounds of ∥f – B͠_n,q,λ(f)∥_∞ are smaller than ∥f – B͠_n,q,0(f)∥_∞, where B͠_n,q,0(f; x) are the classical q-Bernstein operators. So it is meaningful to consider the operators in the form (3).

Data availability statement: The data used to support the findings of this study are included within the article.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 11601266), the Natural Science Foundation of Fujian Province of China (Grant Nos. 2016J05017, 2016J01040), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2018C087R), the Program for New Century Excellent Talents in Fujian Province University, Fujian Provincial Scholarship for Overseas Study and the High-level Personnel of Special Support Program of Xiamen University of Technology (YKJ15030R). We also thank Fujian Provincial Key Laboratory of Data-Intensive Computing, Fujian University Laboratory of Intelligent Computing and Information Processing and Fujian Provincial Big Data Research Institute of Intelligent Manufacturing of China.

References

[1] Lupaş A., A q-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, 1987, 9, 85-92Search in Google Scholar

[2] Phillips G. M., Bernstein polynomials based on the q-integers, Annals of Numerical Mathematics, 1997, 4, 511-518Search in Google Scholar

[3] Nowak G., Approximation properties for generalized q-Bernstein polynomials, Journal of Mathematical Analysis and Applications, 2009, 350, 50-5510.1016/j.jmaa.2008.09.003Search in Google Scholar

[4] Il’inskii A., Convergence of generalized Bernstein polynomial, Journal of Approximation Theory, 2002, 116, 100-11210.1006/jath.2001.3657Search in Google Scholar

[5] Wang H., Meng F., The rate of convergence of q-Bernstein polynomials, Journal of Approximation Theory 2005, 136, 151-15810.1016/j.jat.2005.07.001Search in Google Scholar

[6] Mursaleen M., Khan A., Generalized q-Bernstein-Schurer operators and some approximation theorems, Journal of Function Spaces and Applications, Volume 2013, Article ID 719834, 7 pages, http://dx.doi.org/10.1155/2013/71983410.1155/2013/719834Search in Google Scholar

[7] Gupta V., Finta Z., On certain q-Durrmeyer type operators, Applied Mathematics and Computaion, 2009, 209, 415-42010.1016/j.amc.2008.12.071Search in Google Scholar

[8] Gupta V., Some approximation properties of q-Durrmeyer operators, Applied Mathematics and Computation, 2008, 197, 172-17810.1016/j.amc.2007.07.056Search in Google Scholar

[9] Gupta V., Wang H., The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Mathematical Methods in the Applied Sciences, 2008, 31, 1946-195510.1002/mma.1012Search in Google Scholar

[10] Mursaleen M., Khan F., Khan A., Approximation properties for King’s type modified q-Bernstein-Kantorovich operators, Mathematical Methods in the Applied Sciences, 2015, 36(9), 1178-119710.1002/mma.3454Search in Google Scholar

[11] Agrawal P. N., Finta Z., Kumar A. S., Bernstein-Schurer-Kantorovich operators based on q-integers, Applied Mathematics and Computation, 2015, 256, 222-23110.1016/j.amc.2014.12.106Search in Google Scholar

[12] Dalmanog̃lu Ö., Dog̃ru O., On statistical approximation properties of Kantorovich type q-Bernstein operators, Mathematical and Computer Modelling, 2010, 52, 760-77110.1016/j.mcm.2010.05.005Search in Google Scholar

[13] Cai Q. -B., Lian B. -Y., Zhou G., Approximation properties of λ-Bernstein operators, Journal of Inequalities and Applications, 2018, 10.1186/s13660-018-1653-7Search in Google Scholar PubMed PubMed Central

[14] Gasper G., Rahman M., Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications. Cambridge University press, Cambridge, UK., 35, 1990Search in Google Scholar

[15] Kac V. G., Cheung P., Quantum Calculus, Universitext, Springer-Verlag, New York, 200210.1007/978-1-4613-0071-7Search in Google Scholar

[16] Niven I., Zuckerman H. S., Montgomery H., An Introduction to the Theory of Numbers, 5th edition, Wiley, New York, 1991Search in Google Scholar

[17] Fast H., Sur la convergence statistique, Colloq. Math., 1951, 2, 241-24410.4064/cm-2-3-4-241-244Search in Google Scholar

[18] Gadjiev A. D., Orhan C., Some approximation theorem via statistical convergence, Rocky Mountain J. Math., 2002, 32(1), 129-13810.1216/rmjm/1030539612Search in Google Scholar

[19] Goldman R. N., Pyramid Algorithms: A dynamic programming approach to curves and surfaces for geometric modeling, Elsiver Inc. 2003Search in Google Scholar

Received: 2018-12-31

Accepted: 2019-03-12

Published Online: 2019-05-30

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0039

Keywords for this article

q-integers; λ-Bernstein operators; Statistical convergence; basis function

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