# Approximation Theorem for New Modification ofq-Bernstein Operators on (0,1)

Authors: Yun-Shun Wu; Wen-Tao Cheng; Feng-Lin Chen; Yong-Hui Zhou
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this work, we extend the works of F. Usta and construct new modifiedq-Bernstein operators using the second central moment of the q-Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’s K-functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.


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## Body

## 1. Introduction

In [1], Phillips introduced q-analogue of Bernstein operators as follows:
(1)Blqζ;z=∑i=0lζiqlqpl,iqz,z∈0,1,where pl,iqz=liqzi1−zql−i, i=0,1,⋯,l, and ζ∈C0,1. Later, generalizations of q-Bernstein operators (1) attracted a lot of interest and were constructed and researched widely by a number of researchers. For instance, in [2], Mahmudov and Sabancigil introduced q-Bernstein-Kantorovich operators and studied local and global approximation properties. In [3], Acu et al. defined modified q-Bernstein-Kantorovich operators and established the shape-preserving properties of these operators, e.g., monotonicity and convexity. Some other papers also mention Bernstein operators with parameter(s) and their modification: λ-Bernstein operators [4], α-Bernstein operators [5–8], α,q-Bernstein operators [9], p,q-Bernstein operators [10], p,q-Bernstein-Stancu operators [11], p,q-Bernstein-Kantorovich operators [12], generalized Bernstein operators [13], and so on.In this article, we considerq-analogue of the following new Bernstein operators constructed by the second central moment of the classic Bernstein operators which was given in [14].
(2)Bl∗ζ;z=1l∑i=0llii−lz2zi−11−zl−i−1ζil.There are many papers about the research and application ofq-operators, and we mention some of them: q-Bleimann-Butzer-Hahn operators [15], Bivariater q-Meyer-König-Zeller operators [16], q-Baskakov operators [17, 18], q-Meyer-König-Zeller-Durrmeyer operators [19], q-Phillips operators [20, 21], q-Szász operators [22], q-Bernstein operators [23], and so on. All this achievement motivates us to construct the q-analogue of the operators (2). Before continuing further, let us recall some useful concepts and notations from q-calculus, which can be found in [24]. For nonnegative integer l, the q-integer lq, q-factorial lq!, and q-binomial coefficients liq are defined by
(3)lq=1+q+⋯+ql−1=1−ql1−q,q≠1,l,q=1,lq!=1q2q⋯lq,l≥1,1,l=0,liq=lq!iq!l−iq!,i=0,1,⋯,l.Further,q-power basis can be defined by
(4)x−yql=x−yx−qy⋯x−ql−1y,l=1,2,⋯,1,l=0,x−q−1y⋯x−qly,l=−1,−2,⋯.Theq-derivative Dqζ of a function ζ can be defined by
(5)Dqζz=ζz−ζqz1−qz,ifz≠0,and Dqζ0=ζ′0 provided ζ′0 exists. High-order q-derivatives can be defined by Dq0ζ=ζ, Dqnζ=DqDqn−1ζ, n=1,2,⋯. The formula for the q-derivative of a product is
(6)Dqζxηx=Dqζxηx+Dqηxζqx.Theq-analogue of new-Bernstein operators (2) on 0,1 is defined by the following:
(7)Blqζ;z=1lq∑i=0lliqzi−11−qzql−i−1iq−lqz2ζiqlq,where q∈0,1, l∈ℕ+≔1,2,⋯, z∈0,1, and ζ∈C0,1.The rest of the paper is organized as follows: In Section2, we get the basic results by the moment computation formulas. And the first, second, fourth, and sixth order central moment computation formulas and limit equalities are also computed. In Sections 3 and 4, we investigate the Korovkin-type approximation theorem and the Voronovskaja-type asymptotic formula for the operators (7). In Section 5, we obtain two local approximation theorems using Peetre’s K-functional and Steklov mean and in terms of modulus of smoothness. In Section 6, we study the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators (7). In Section 7, we show some graphs and numerical examples to analyze the theoretical results by using Matlab algorithms.


## 2. Auxiliary Lemmas

In this section, we present certain auxiliary results which will be used to prove our main theorems for the operators (7). Using the lemma in ([25], Lemma 2), we have the moment computation formulas for the operators (1):Lemma 1.
If we defineTl,mqz=Blqum;z, then there holds the following relation:
(8)Tl,0qz=1,Tl,1qz=z,Tl,2qz=z2+z1−zlq,(9)lqTl,m+1qz=z1−zDqTl,mqz+lqzTl,mqz,m=0,1,2,⋯.Now, we give the moment relation between the operators (1) and the operators (7) as follows:Lemma 2.
If we defineTl,mqz=Blqum;z, then there holds the following relation:
(10)z1−zlqTl,mqz=Tl,m+2qz−2zTl,m+1qz+z2Tl,mqz,(11)Tl,mqz=DqTl,m+1qz−zDqTl,mqz,(12)Tl,mqz=1lqDqz1−zDqTl,mqz+Tl,mqqz,where q∈0,1, l∈ℕ+≔1,2,⋯, m∈ℕ≔0,1,2,⋯, and z∈0,1.Proof.
By the definition ofBlq and Blq, we can obtain
(13)z1−zlqTl,mqz=z1−zlq2∑i=0lliqzi−11−qzql−i−1iq−lqz2iqlqm=∑i=0lliqzi1−zql−iiqlq−z2iqlqm=Tl,m+2qz−2zTl,m+1qz+z2Tl,mqz.
Next, by (9) and (10), we can obtain
(14)z1−zTl,mqz=lqTl,m+2qz−2lqzTl,m+1qz+lqz2Tl,mqz=z1−zDqTl,m+1qz−lqzTl,m+1qz+lqz2Tl,mqz=z1−zDqTl,m+1qz−zlqTl,m+1qz+lqzTl,mqz=z1−zDqTl,m+1qz−z21−zDqTl,mqz.
Thus, we complete the proof of (11). Finally, by (6), (9), and (11), we can obtain
(15)Tl,mqz=DqTl,m+1qz−zDqTl,mqz=Dqz1−zlqDqTl,mqz+zTl,mqz−zDqTl,mqz=1lqDqz1−zDqTl,mqz+DqzTl,mqz−zDqTl,mqz=1lqDqz1−zDqTl,mqz+Tl,mqqz.
We complete the proof of the Lemma2. ☐Then, the following lemma can be obtain immediately:Lemma 3.
Forq∈0,1, l∈ℕ+, and z∈0,1, we have
(16)Tl,0qz=1,Tl,1qz=q−2qlqz+1lq,Tl,2qz=q2−q2+2q3qlq+2q3qlq2z2+q+2q2lq−2q2+2qlq2z+1lq2,αlqz≔Blqu−z;z=q−1z+1−2qzlq,βlqz≔Blqu−z2;z=q−12−q3+3q2−1lq+2q3qlq2z2+q2+3q−1l−2q2+2qlq2z+1lq2.Lemma 4.
The sequenceql satisfies ql∈0,1, such that ql⟶1 and qll⟶k∈0,1 as l⟶∞; then, for any z∈0,1, we have
(17)liml⟶∞lqlαlqlz=1+k−3z,(18)liml⟶∞lqlβlqlz=3z1−z,(19)liml⟶∞lqlBlqlu−z4;z=0.Proof.
First, we prove the limitliml⟶∞lql=+∞. In fact, for any q∈0,1, such sufficiently large l0∈ℕ+ that l0q>1/21−q. But for l>l0 such that ql>q, we easily have lql>l0q. Applying Lemma 3, we can directly obtain (17) and (18). As l⟶∞, using Lemma 1, we can rewrite
(20)Tl,3qlz=z3+2ql+1lqlz21−z+o1lql,Tl,4qlz=z4+3ql+2ql+1lqlz31−z+o1lql.
Applying (12), we can obtain
(21)Tl,2qlz=ql2−ql2+2ql3qllqlz2+ql+2ql2lqlz+o1lql,Tl,3qlz=ql3−ql31+2ql+3ql4qllqlz3+3ql2+ql21+2qllqlz2+o1lql,Tl,4qlz=ql4−ql41+2ql+3ql+4ql5qllqlz4+4ql2+ql31+2ql+3qllqlz3+o1lql.
Combining Lemma3 and Blqlu−z4;z=∑m=044m−14−mBlqlu4−m;zzm, we can obtain
(22)Blqlu−z4;z=ql−14z4−A4ql−4A3ql+6A2ql−4A1qllqlz4+B4ql−4B3ql+6B2ql−4B1qllqlz3+o1lql,where A4ql=ql41+2ql+3ql+4ql5ql, A3ql=ql31+2ql+3ql4ql, A2ql=ql2+2ql3ql, A1ql=2ql, B4ql=4ql2+ql31+2ql+3ql, B3ql=3ql2+ql21+2ql, B2ql=2ql2+ql, and B1ql=1. Hence, A4ql~26, A3ql~15, A2ql~7, A1ql~2, B4ql~22, B3ql~12, B2ql~5, and B1ql~1 as l⟶∞, we have
(23)liml⟶∞A4ql−4A3ql+6A2ql−4A1ql=26−4×15+6×7−4×2=0,liml⟶∞B4ql−4B3ql+6B2ql−4B1ql=22−4×12+6×5−4×1=0.Combining liml⟶∞lqlql−14=0, we can obtain (19). ☐Lemma 5.
Forq∈0,1, l∈ℕ+, and ζ∈CB0,1, we can have ∥Blqlζ;z∥≤∥ζ∥, where CB0,1 denotes the set of all real-valued bounded and continuous functions defined on 0,1, endowed with the norm ζ=supz∈0,1ζz.Proof.
In view of (7) and Lemma 3, for any z∈0,1, we have
(24)Blqlζ;z≤Tl,0qzζ=ζ.
Taking supremum over allz∈0,1, we obtain the required result. ☐


## 3. Korovkin Approximation Theorem

Theorem 6.
Let the sequenceql satisfy ql∈0,1, for any ζ∈CB0,1; then, the sequence Blqlζ;z converges uniformly to ζ on 0,1 if and only if ql⟶1 as l⟶∞.Proof.
Letql∈0,1 and ql⟶1 as l⟶∞; then, we have lql⟶∞ as l⟶∞. By the Korovkin theorem ([26], p. 8, Theorem 6), it is sufficient to prove the three following limit equalities:
(25)liml⟶∞Blqlum;z−zm=0,m=0,1,2.
We can obtain these three limit equalities easily by Lemma3. Thus, we get that the sequence Blqlζ;z converges uniformly to ζ on 0,1.
We prove the converse result by contradiction. Assume thatql does not tend to 1 as l⟶∞; then, it must contain a subsequence qlm⊂0,1 with lm≥1, such that qlm⟶q∈0,1. Thus,
(26)limm⟶∞1lmqlm=limm⟶∞1−qlm1−qlmlm=1−q.
Takingl=lm and q=qlm in Blqu;z and using Lemma 3, we can obtain
(27)Blmqlmu;z=Tlm,1qlmz⟶1−21−qz+1−q≠z,as m⟶∞. This leads to a contradiction; hence, ql⟶1 as l⟶∞. The proof is completed. ☐


## 4. Voronovskaja-Type Theorem

In this section, we give a Voronovskaja-type asymptotic formula for the operators (7) by means of the first, second, and fourth central moments.Theorem 7.
Under the condition of Lemma4 and ζ∈CB0,1. Suppose that ζ′′z exists at a point z∈0,1; then, we have
(28)liml⟶∞lqlBlqlζ;z−ζz=1+k−3zζ′z+32z1−zζ′′z.Proof.
Applying the Taylor’s expansion formula forζ, we have
(29)ζu=ζz+ζ′zu−z+12ζ′′zu−z2+Θu;zu−z2,where
(30)Θu;z=ζu−ζz−ζ′zu−z−1/2ζ′′zu−z2u−z2,u≠z,0,u=z.
Using the L’Hospital’s Rule,(31)limu⟶zΘu;z=12limu⟶zζ′u−ζ′zu−z−12ζ′′z=0.
Thus,Θ.;z∈CB0,1. Applying Blql to both sides of (29) and using Lemma 3, we have
(32)Blqlζ;z−ζz=ζ′zαlqlz+12ζ′′zβlqlz+BlqlΘu;zu−z2;z.
Applying the Cauchy-Schwarz inequality, we have(33)BlqlΘu;zu−z2;z≤BlqlΘ2u;z;zBlqlu−z4;z.
By Theorem6, we can obtain
(34)liml⟶∞Θ2u;z=Θ2z;z=0.
From (19), (33), and (34), we have
(35)liml⟶∞lqlBlqlΘu;zu−z2;z=0.
Combining (17), (18), and (35), we complete the proof of Theorem 7. ☐Corollary 8.
Under the condition of Lemma4 and ζ′′∈CB0,1, then
(36)liml⟶∞lqlBlqlζ;z−ζz=1+k−3zζ′z+32z1−zζ′′z,uniformly in z∈0,1.


## 5. Local Approximation

Firstly, we recall the first and second order modulus of continuity ofζ∈CB0,1 are defined, respectively, by
(37)ωζ;δ=sup0<t≤δsupz,z+t∈0,1ζz+t−ζz,ω2ζ;δ=sup0<t≤δsupz,z+2t∈0,1ζz+2t−2ζz+t+ζz.The Peetre’sK-functional is defined by
(38)Kζ;δ=infη,η′′∈CB0,1ζ−η+δη′′.By ([26], p. 177, Theorem 2.4), there exists an absolute constant C>0 depending only on ζ such that
(39)Kζ;δ≤Cω2ζ;δ.Theorem 9.
Under the condition of Lemma4, then for all ζ∈CB0,1 and z∈0,1, there exists an absolute positive constant C1=4C such that
(40)Blqlζ;z−ζz≤C1ω2ζ;αlqlz2+∣βlqlz∣+ωζ;αlqlz.Proof.
Forζ∈CB0,1, we define new operators by
(41)B¯lqlζ;z=Blqlζ;z+ζz−ζαlqlz+z.
By Lemma3, we can obtain immediately
(42)B¯lql1;z=Blql1;z=Tl,0qlz=1,B¯lql1;z=Blqlu;z+z−αlqlz+z=Tl,1qlz−αlqlz=z.
For anyη,η′′∈CB0,1 and u,z∈0,1, by the Taylor’s expansion formula, we can obtain
(43)ηu=ηz+η′zu−z+∫zuu−vη′′vdv.
Applying the operators (7) to both sides of the above equation, we have
(44)B¯lqlη;z=ηz+B¯lql∫zuu−vη′′vdv;z=ηz+Blql∫zuu−vη′′vdv;z−∫zαlqlz+zαlqlz+z−vη′′vdv.
Hence,(45)B¯lqlη;z−ηz=B¯lql∫zuu−vη′′vdv;z≤Blql∫zuu−vη′′vdv;z+∫zαlqlz+zαlqlz+z−vη′′vdv≤βlqlz+αlqlz2η′′.
By Lemma5, we have
(46)B¯lqlζ;z≤Blqlζ;z+ζz+ζαlqlz+z≤3ζ,∀ζ∈CB0,1.
For anyζ∈CB0,1 and η,η′′∈CB0,1, combining (45) and (46), we obtain
(47)Blqlζ;z−ζz=B¯lqlζ;z−ζz+ζαlqlz+z−ζz≤B¯lqlζ−η;z+B¯lqlη;z−ηz+ηz−ζz+ζαlqlz+z−ζz≤4ζ−η+βlqlz+αlqlz2η′′+ωζ;αlqlz.
Taking infinum on the right hand side over allη′′∈CB0,1, using (39), we obtain the desired assertion. ☐Now, we discuss local approximation theorems for the operators (7) by Steklov mean. For any ζ∈CB0,1 and h>0, the Steklov mean is defined by
(48)ζhz=4h2∫0h/2∫0h/22ζz+u+v−ζz+2u+2vdudv.In direct computation, it is proved that(49)ζh−ζ≤ω2ζ;h,(50)ζh′,ζh″∈CB0,1,ζh′≤5hωζ;h,ζ′′h≤9h2ω2ζ;h,while ζ∈CB0,1.Theorem 10.
Under the condition of Lemma4, then for all ζ∈CB0,1 and z∈0,1,
(51)Blqlζ;z−ζz≤5lqlαlqlzωζ;1lql+92lqlβlqlz+2ω2ζ;1lql.Proof.
Forz,h,z+2h∈0,1, by the definition of the Steklov mean, we can write
(52)Blqlζ;z−ζz≤Blqlζ−ζh;z+Blqlζh;z−ζhz+ζhz−ζz.
By Lemma5 and (49), we have
(53)Blqlζ−ζh;z≤Blqlζ−ζh;z≤ζ−ζh≤ω2ζ;h.
Now, by the Taylor’s expansion formula forζh, we have
(54)ζhu=ζhz+ζh′zu−z+∫zuu−vζ′′hvdv.
Combining (49) and (50), we have
(55)Blqlζh;z−ζhz≤Blqlζh′zu−z;z+Blql∫zuu−vζ′′hvdv;z≤ζh′Blqlu−z;z+ζh″Blql∫zuu−vdv;z≤ζh′αlqlz+12ζh″βlqlz≤5hαlqlzωζ;h+92h2βlqlzω2ζ;h.
Hence,(56)Blqlζ;z−ζz≤5hωζ;hαlqlz+92h2βlqlz+2ω2ζ;h,for z∈0,1. Setting h=1/lql, we obtain the desired result. ☐


## 6. Rate of Convergence

First, we discuss the rate of convergence of the operators (7) by means of the modulus of continuity ωζ;δ.Theorem 11.
Under the condition of Lemma4, then for all ζ∈C0,1 and z∈0,1,
(57)Blqlζ;z−ζz≤2ωζ;βlqlz.Proof.
Using ([26], p. 41, (6.5)), for all ζ∈C0,1 and δ>0, we have
(58)ζu−ζz≤ωζ;δ1+u−zδ.
Applying the monotonicity and the linearity ofBlql and Cauchy-Schwarz inequality, for any δ>0, we have
(59)Blqlζ;z−ζz≤Blqlζu−ζz;z≤Blqlωζ;u−z;z≤ωζ;δ1+Blqlu−z;zδ≤ωζ;δ1+Blqlu−z2;zδ≤ωζ;δ1+βlqlzδ,by taking δ=βlqlz. We complete the proof of Theorem 11. ☐Theorem 12.
Under the condition of Lemma4, then for all ζ′∈C0,1 and z∈0,1,
(60)Blqlζ;z−ζz≤αlqlz+2βlqlzωζ′;βlqlz.Proof.
ApplyingBlql to both sides of ζu=ζz+ζ′zu−z+ζu−ζz−ζ′zu−z, we have
(61)Blqlζ;z−ζz≤ζ′zBlqlu−z;z+Blqlζu−ζz−ζ′zu−z;z≤αlqlzζ′z+Blqlu−z1+u−zδ;zωζ′;δ≤αlqlzζ′z+Blqlu−z2;z1+Blqlu−z2;zδωζ′;δ,with the help of Cauchy-Schwartz inequality and mean value theorem. Taking δ=Blqlu−z2;z and by Lemma 3, we can get the desired result. ☐Next, we discuss the rate of convergence of the operators (7) by means of three Lipschitz classes: LipMγ, LipMs,tγ, and LipMγ,I. A function ζ∈C0,1 belongs to LipMγγ∈0,1, if the condition
(62)ζu−ζz≤Mu−zγ,u,z∈0,1,is satisfied, where M is a positive constant depending only on γ and ζ.Theorem 13.
Under the condition of Lemma4, then for all ζ∈LipMγ and z∈0,1,
(63)Blqlζ;z−ζz≤Mβlqlzγ/2.Proof.
According to the monotonicity and the linearity of the operators (7) and taking into account that ζ∈LipMγ, we can obtain
(64)Blqlζ;z−ζz≤Blqlζu−ζz;z≤MBlqlu−zγ;z.
Applying well-known Hölder’s inequality witht1=2/γ and t2=2/2−γ, we can get
(65)Blqlζ;z−ζz≤MBlqlu−zγ;z≤MBlqlu−zt1γ;x1/t1Blql1t2;x1/t2=MBlqlu−t2;xγ/2=Mβlqlzγ/2.
We obtain the required result. ☐In [27], Özarslan and Aktug˘lu constructed the following Lipschitz-type space LipMs,tγ with two distinct parameters s,t>0 as follows:
(66)LipMs,tγ≔ζ∈C0,1:ζu−ζz≤Mu−zγu+sz+tz2,u,z∈0,1,where γ∈0,1 and M is a positive constant depending only on γ,s,t, and ζ.Theorem 14.
Under the condition of Lemma4, then for all ζ∈LipMs,tγ and z∈0,1,
(67)Blqlζ;z−ζz≤Mβlqlzsz+tz2γ/2.Proof.
Applying the well-known Hölder inequality withk1=2/γ and k2=2/2−γ, we have
(68)Blqlζ;z−ζz≤Blqlζu−ζz;z≤BlqlMu−zγu+sz+tz2;z≤Msz+tz2Blqlu−zγ;z≤Msz+tz2Blqlu−zγk1;z1/k1Blql1k2;z1/k2=Mβlqlzsz+tz2γ/2.
Thus, the proof of Theorem14 is completed. ☐A functionζ∈C0,1 belongs to LipMγ,Iγ∈0,1,I⊂0,1, if the condition
(69)ζu−ζz≤Mu−zγ,u∈I,z∈0,1,is satisfied, where M is a positive constant depending only on γ and ζ.Theorem 15.
Under the condition of Lemma4, then for all ζ∈C0,1⋂LipMγ,I and z∈0,1,
(70)Blqlζ;z−ζz≤Mβlqlzγ/2+2dγz;I,where dz;I=infu−z:u∈I denotes the distance between z and I.Proof.
LetI¯ be the closure of I. Using the properties of infimum, there is at least a point u0∈I¯ such that dz;I=z−u0. By the triangle inequality
(71)ζu−ζz≤ζu−ζu0+ζz−ζu0,we have
(72)Blqlζ;z−ζz≤Blqlζu−ζu0;z+Blqlζz−ζu0;z≤MBlqlu−u0γ;z+z−u0γ≤MBlqlu−zγ+z−u0γ;z+z−u0γ≤MBlqlu−zγ;z+2z−u0γ.
Choosingk1=2/γ and k2=2/2−γ and using the well-known Hölder inequality, we have
(73)Blqlζ;z−ζz≤MBlqlu−zγk1;z1/k1Blql1k2;z1/k2+2dγz;I≤MBlqlu−z2;z1/k1+2dγz;I≤Mβlqlzγ/2+2dγz;I.
This completes the proof. ☐


## 7. Numerical Examples

In this section, we will analyze the theoretical results presented in the previous sections by numerical examples.Letζz=1−cos4ez, z∈0.01,0.99, ql=1−1/l2, and l∈10,20,50,100. The convergence of the operators Blql to function ζ is shown in Figure 1. The error of approximation Blqlζ;z−ζz is given in Figure 2. Meantime, we compute the error of approximation Blqlζ;z−ζz while l=10,20,50,100 at points 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 in Table 1.Figure 1
Approximation process byBlql.Figure 2
Error of approximationBlqlζ;z−ζz.Table 1
Error of approximationBlqlζ;z−ζz for l=10,20,50,100.
zB10q10ζ;z−ζzB20q20ζ;z−ζzB50q50ζ;z−ζzB100q100ζ;z−ζz0.10.37530.21560.09160.04650.20.05360.08340.04960.02770.30.44810.20300.06880.03160.40.87970.51070.21770.11040.50.95270.62370.29320.15460.60.48220.36580.18750.10250.70.43190.20940.09110.04670.81.27250.72150.31780.16930.91.17030.71940.21190.0921



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