Generalized 𝑞-Bernstein-Schurer Operators and Some Approximation Theorems

Abstract

We study statistical approximation properties of 𝑞-Bernstein-Shurer operators and establish some direct theorems. Furthermore, we compute error estimation and show graphically the convergence for a function 𝑓 by operators and give its algorithm.

1. Introduction and Preliminaries

In 1987, Lupaş [1] introduced the first 𝑞-analogue of Bernstein operator and investigated its approximating and shape-preserving properties. Another 𝑞-generalization of the classical Bernstein polynomials is due to Phillips [2]. After that many generalizations of well-known positive linear operators, based on 𝑞-integers, were introduced and studied by several authors. Recently the statistical approximation properties have also been investigated for 𝑞-analogue polynomials. For instance, in [1] 𝑞-analogues of Bernstein-Kantorovich operators; in [3] 𝑞-Baskakov-Kantorovich operators; in [4] 𝑞-Szász-Mirakjan operators; in [5, 6] 𝑞-Bleimann, Butzer and Hahn operators; in [7] 𝑞-analogue of Baskakov and Baskakov-Kantorovich operators; in [8] 𝑞-analogue of Szász Kantorovich operators; in [9, 10] 𝑞-analogue of Stancu-Beta operators; and in [11] 𝑞-Lagrange polynomials were defined and their classical approximation or statistical approximation properties were investigated.

Schurer [12] introduced the following operators 𝐿_𝑚,𝑝∶𝐶[0,𝑝+1]→𝐶[0,1] defined for any 𝑚∈ℕ and any function 𝑓∈𝐶[0,𝑝+1]: 𝐿_𝑚,𝑝𝑓;𝑥=^𝑚+𝑝_𝑘=0𝑘𝑥𝑚+𝑝_𝑘(1−𝑥)^{𝑚+𝑝−𝑘}𝑓𝑘𝑚,𝑥∈[0,1]. (1)

Recently, Muraru [13] introduced the 𝑞-analogue of these operators and investigated their approximation properties and rate of convergence using modulus of continuity. Note that Radu [14] has also used 𝑞-intgers to define and study the approximation properties of the 𝑞-analogue of Kantorovich operators.

In this paper, we study the statistical approximation properties by 𝑞-Bernstein-Schurer operators. We also give some direct theorems.

We recall certain notations of 𝑞-calculus. Let 𝑞>0. For any 𝑘∈ℕ₀∶={0}∪ℕ, the 𝑞-integer [𝑘]_𝑞 is defined by [𝑘]_𝑞∶=1−𝑞^𝑘1−𝑞,𝑞≠1,𝑘,𝑞=1, (2) and the 𝑞-factorial [𝑘]_𝑞! by [𝑘]_𝑞[𝑘]!∶=^𝑞[𝑘−1]_𝑞⋯[1]_𝑞,𝑘≥1,1,𝑘=0. (3) Also, the 𝑞-binomial coefficients are defined by 𝑛𝑘_𝑞[𝑛]∶=^𝑞![𝑘]_𝑞![𝑛−𝑘]_𝑞!. (4) Details on 𝑞-integers can be found in [14].

Muraru [13] introduced the following operators known as the generalized 𝑞-Bernstein-Schurer operators. For any 𝑚∈ℕ, 𝑝 a fixed positive integer, and 𝑓∈𝐶[0,𝑝+1], 𝐿^∗_𝑚,𝑝=𝑓;𝑞;𝑥_{𝑚+𝑝𝑘=0}𝑘𝑚+𝑝_𝑞𝑥^{𝑘𝑚+𝑝−𝑘−1}_𝑠=01−𝑞^𝑠𝑥𝑓[𝑘]_𝑞[𝑚]_𝑞,𝑥∈[0,1]. (5)

We note the following properties as in [13] for 𝐿^∗_𝑚,𝑝(𝑓;𝑞;𝑥).

Lemma 1. For 𝑥∈[0,1], 0<𝑞≤1, 𝐿^∗_𝑚,𝑝1;𝑞;𝑥=1. (6)

Lemma 2. For 𝑥∈[0,1], 0<𝑞≤1, 𝐿^∗_𝑚,𝑝𝑡;𝑞;𝑥=𝑥𝑚+𝑝_𝑞[𝑚]_𝑞. (7)

Lemma 3. For 𝑥∈[0,1], 0<𝑞≤1, 𝐿^∗_𝑚,𝑝𝑡²;𝑞;𝑥=𝑚+𝑝_𝑞[𝑚]²_𝑞𝑚+𝑞_𝑞𝑥²+𝑥(1−𝑥). (8)

Lemma 4. Let 𝑞∈(0,1). Then for 𝑥∈[0,1]𝐿^∗_𝑚,𝑝𝑡−𝑥;𝑞;𝑥=𝑥𝑚+𝑝_𝑞[𝑚]_𝑞−1. (9)

Lemma 5. Let 𝑞∈(0,1). Then for 𝑥∈[0,1]𝐿^∗_𝑚,𝑝(𝑡−𝑥)²;𝑞;𝑥=𝑥²𝑚+𝑝_𝑞−[𝑚]_𝑞²−𝑚+𝑝_𝑞[𝑚]²_𝑞+𝑥𝑚+𝑝_𝑞[𝑚]²_𝑞. (10)

2. Statistical Approximation

In this section we obtain the Korovkin type weighted statistical approximation properties for these operators defined in (5). Korovkin type approximation theory [15] has also many useful connections, other than classical approximation theory, in other branches of mathematics (see Altomare and Campiti [16]).

First we recall the concept of statistical convergence for sequences of real numbers which was introduced by Fast [17] and further studied by many others.

Let 𝐾⊆ℕ and 𝐾_𝑛={𝑗≤𝑛∶𝑗∈𝐾}. Then the natural density of 𝐾 is defined by 𝛿(𝐾)=lim_𝑛𝑛⁻¹|𝐾_𝑛| if the limit exists, where |𝐾_𝑛| denotes the cardinality of the set 𝐾_𝑛.

A sequence 𝑥=(𝑥_𝑗) of real numbers is said to be 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑎𝑙𝑙𝑦𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡 to 𝐿 provided that for every 𝜖>0 the set {𝑗∈ℕ∶|𝑥_𝑗−𝐿|≥𝜖} has natural density zero; that is, for each 𝜖>0, lim_𝑛1𝑛𝑗≤𝑛∶𝑥_𝑗−𝐿≥𝜖=0. (11)

In this case, we write 𝑠𝑡−lim_𝑛𝑥_𝑛=𝐿. Note that every convergent sequence is statistically convergent but not conversely, even unbounded sequence may be statistically convergent. For example, let 𝑢=(𝑢_𝑚) be defined by 𝑢_𝑚=1,if𝑘isasquare,0,otherwise. (12)

Then 𝑠𝑡−lim𝑢_𝑚=0, but 𝑢 is not convergent.

Recently the idea of statistical convergence has been used in proving some approximation theorems, in particular, Korovkin type approximation theorems by various authors, and it was found that the statistical versions are stronger than the classical ones. Authors have used many types of classical operators and test functions to study the Korovkin type approximation theorems which further motivate continuation of this study. After the paper of Gadjiev and Orhan [18], different types of summability methods have been deployed in approximation process, for example, [19–23]. Recently, 𝐴-statistical convergence has been used to the summability of Walsh-Fourier series [24].

Let 𝐶_𝐵[0,𝑝+1] be the space of all bounded and continuous functions on [0,𝑝+1]. Then 𝐶_𝐵[0,𝑝+1] is a normed linear space with ‖𝑓‖=sup_𝑥≥0|𝑓(𝑥)|. Let 𝑤 be a function of the type of modulus of continuity. The principal properties of the function are the following:(i)𝑤 is a nonnegative increasing function on [0,𝑝+1],(ii)lim_𝛿→0𝑤(𝛿)=0.

Let 𝐶_𝐵[0,𝑝+1] be the space of all real valued functions 𝑓 defined on [0,𝑝+1] satisfying the following condition: 𝑓(𝑥)−𝑓𝑦≤𝑤𝑥−𝑦, (13) for any 𝑥,𝑦∈[0,𝑝+1].

We consider a sequence 𝑞=(𝑞_𝑛), 𝑞_𝑛∈(0,1), such that lim_𝑛→∞𝑞_𝑛=1. (14) The condition (14) guarantees that [𝑛]_𝑞𝑛→∞ as 𝑛→∞.

Now our first result is as follows.

Theorem 6. Let (𝐿^∗_𝑚,𝑝) be the sequence of the operators (5), and the sequence 𝑞=(𝑞_𝑚) satisfies (14). Then for any function 𝑓∈𝐶_𝐵[0,𝑝+1], 𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝𝑓;𝑞_𝑚;𝑥−𝑓=0. (15)

Proof. Let 𝑒_𝜈=𝑥^𝜈 where 𝜈=0,1,2. Since 𝐿^∗_𝑚,𝑝(1;𝑞_𝑚,𝑥)=𝑞_𝑚, therefore we can write 𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚1;𝑞_𝑚;𝑥−1=𝑠𝑡−lim_𝑛→∞𝑒₀𝑞_𝑚−1 (16) as 𝐿^∗_𝑚,𝑝1;𝑞_𝑚;𝑥−1=𝐿^∗_𝑚1;𝑞_𝑚=𝑒;𝑥−1₀𝑞_𝑚−1≤𝑞_𝑚−1. (17) By (14), it can be observed that 𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝1;𝑞_𝑚;𝑥−1=0. (18) Similarly 𝐿^∗_𝑚,𝑝𝑡;𝑞_𝑚;𝑥−𝑥=𝑒₂𝑞_𝑚𝑚+𝑝²_𝑞𝑚[𝑚]²_𝑞𝑚≤−1𝑚+𝑝_𝑞[𝑚]_𝑞−1. (19) For a given 𝜖>0, let us define the following sets: 𝑈=𝑚∶𝐿^∗_𝑚,𝑝𝑡;𝑞_𝑚,𝑈;𝑥−𝑥≥𝜖_′=𝑚∶𝑚+𝑝_𝑞𝑚[𝑚]_𝑞𝑚−1≥𝜖. (20) It is obvious that 𝑈⊆𝑈^′; it can be written as 𝛿𝑘≤𝑚∶𝐿^∗_𝑚,𝑝𝑡;𝑞_𝑚;𝑥−𝑥≥𝜖≤𝛿𝑘≤𝑚∶𝑚+𝑝_𝑞𝑚[𝑚]_𝑞𝑚−1≥𝜖. (21) By using (14), we get 𝑠𝑡−lim_𝑚→∞𝑚+𝑝_𝑞𝑚[𝑚]_𝑞𝑚−1=0. (22) Therefore 𝛿𝑘≤𝑚∶𝑚+𝑝_𝑞𝑚[𝑚]_𝑞𝑚−1≥𝜖=0, (23) and we have 𝑠𝑡−lim_𝑛→∞𝐿^∗_𝑚,𝑝𝑡;𝑞_𝑚;𝑥−𝑥=0. (24) Lastly, we have 𝐿^∗_𝑚,𝑝𝑡²;𝑞_𝑚;𝑥−𝑥²=𝑒₂𝑞_𝑚𝑚+𝑝²_𝑞𝑚[𝑚]²_𝑞𝑛−𝑞𝑚+𝑝_𝑞𝑚[𝑚]²_𝑞𝑚+𝑒−1₁𝑞_𝑚𝑚+𝑝_𝑞𝑚[𝑚]²_𝑞𝑚≤𝑞_𝑚𝑚+𝑝²_𝑞[𝑚]²_𝑞−1. (25) Therefore by (25), we get 𝐿^∗_𝑚,𝑝𝑡²;𝑞_𝑚;𝑥−𝑥²≤1𝑞²_𝑚𝑚+𝑝+1²_𝑞𝑚[𝑚]²_𝑞𝑚+1−1𝑞²_𝑚𝑞_𝑚+2𝑚+𝑝+1_𝑞[𝑚]²_𝑞+1𝑞²_𝑚1+𝑞_𝑚[𝑚]²_𝑞. (26) Now, if we choose 𝛼_𝑚=1𝑞²_𝑚𝑚+𝑝+1²_𝑞𝑚[𝑚]²_𝑞𝑚𝛽−1,_𝑚=𝑞_𝑚+2𝑞²_𝑚𝑚+𝑝+1_𝑞[𝑚]²_𝑞,𝛾_𝑚=1𝑞²_𝑚1+𝑞_𝑚[𝑚]²_𝑞, (27) then, by (14), we can write 𝑠𝑡−lim_𝑚→∞𝛼_𝑚=0=𝑠𝑡−lim_𝑚→∞𝛽_𝑚=𝑠𝑡−lim_𝑚→∞𝛾_𝑚. (28) Now for given 𝜖>0, we define the following four sets: 𝑈=𝑚∶𝐿^∗_𝑚,𝑝𝑡²;𝑞_𝑛;𝑥−𝑥²,𝑈≥𝜖₁=𝑚∶𝛼_𝑚≥𝜖3,𝑈₂=𝑚∶𝛽_𝑚≥𝜖3,𝑈₃=𝑚∶𝛾_𝑚≥𝜖3. (29) It is obvious that 𝑈⊆𝑈₁∪𝑈₂∪𝑈₃. Then we obtain 𝛿𝑘≤𝑚∶𝐿^∗_𝑚,𝑝𝑡²;𝑞_𝑚;𝑥−𝑥²≥𝜖≤𝛿𝑘≤𝑚∶𝛼_𝑚≥𝜖3+𝛿𝑘≤𝑚∶𝛽_𝑚≥𝜖3+𝛿𝑘≤𝑚∶𝛾_𝑚≥𝜖3. (30) Using (28), we get 𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝𝑡²;𝑞_𝑚;𝑥−𝑥²=0. (31) Since 𝐿^∗_𝑚,𝑝𝑓;𝑞_𝑚≤𝐿;𝑥−𝑓_∗𝑚,𝑝𝑡²;𝑞_𝑚;𝑥−𝑥²+𝐿^∗_𝑚,𝑝𝑡;𝑞_𝑚+𝐿;𝑥−𝑥_∗𝑚,𝑝1;𝑞_𝑚;𝑥−1, (32) we get 𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝𝑓;𝑞_𝑚;𝑥−𝑓≤𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝𝑡²;𝑞_𝑚;𝑥−𝑥²+𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝𝑡;𝑞_𝑚;𝑥−𝑥+𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝1;𝑞_𝑚;𝑥−1, (33) which implies that 𝑠𝑡−lim_𝑚→∞𝐿^∗_𝑚,𝑝𝑓;𝑞_𝑚;𝑥−𝑓=0. (34)
This completes the proof of the theorem.

Remark 7. In the following example, we demonstrate that the statistical version is stronger than the ordinary approximation. Let us write 𝑇^∗_𝑚,𝑝(𝑓;𝑞_𝑚;𝑥)=(1+𝑢_𝑚)𝐿^∗_𝑚,𝑝(𝑓;𝑞_𝑚;𝑥), where the sequence (𝑢_𝑚) is defined by (12). Then under the hypothesis of the previous theorem, we have 𝑠𝑡−lim_𝑚→∞𝑇^∗_𝑚,𝑝𝑓;𝑞_𝑚;𝑥−𝑓=0. (35) However, lim_𝑚→∞‖𝑇^∗_𝑚,𝑝(𝑓;𝑞_𝑚;𝑥)−𝑓‖ does not exist, since (𝑢_𝑚) is statistically convergent but not convergent.

3. Direct Theorems

The Peetre’s 𝐾-functional is defined by 𝐾₂𝑓,𝛿=inf𝑓−𝑔+𝛿𝑔^′′∶𝑔∈𝑊²_𝑝, (36) where 𝑊²_𝑝=𝑔∈𝐶_𝐵0,𝑝+1∶𝑔^′,𝑔^′′∈𝐶_𝐵0,𝑝+1. (37)

By [25], there exists a positive constant 𝐶>0 such that 𝐾₂(𝑓,𝛿)≤𝐶𝑤₂(𝑓,𝛿^1/2), 𝛿>0, where the second-order modulus of continuity is given by 𝑤₂𝑓,𝛿^1/2=sup_{0<ℎ≤𝛿1/2}sup_{𝑥∈[0,𝑝+1]}𝑓(𝑥+2ℎ)−2𝑓(𝑥+ℎ)+𝑓(𝑥). (38)

Also for 𝑓∈𝐶_𝐵[0,𝑝+1] the usual modulus of continuity is given by 𝑤𝑓,𝛿=sup_{0<ℎ≤𝛿1/2}sup_{𝑥∈[0,𝑝+1]}𝑓(𝑥+ℎ)−𝑓(𝑥). (39)

Theorem 8. Let 𝑓∈𝐶_𝐵[0,𝑝+1] and 0<𝑞_𝑚<1 such that 𝑞_𝑚→1(𝑚→∞). Then for all 𝑛∈ℕ and 𝑝 fixed, there exists an absolute constant 𝐶>0 such that 𝐿^∗_𝑚,𝑝𝑓;𝑞_𝑚;𝑥−𝑓(𝑥)≤𝐶𝑤₂𝑓,𝛿_𝑚(𝑥), (40) where 𝛿²_𝑚(𝑥)=𝑥²𝑚+𝑝_𝑞−[𝑚]_𝑞²−𝑚+𝑝_𝑞[𝑚]²_𝑞+𝑥𝑚+𝑝_𝑞[𝑚]²_𝑞. (41)

Proof. Let 𝑔∈𝑊₂. From Taylor’s expansion, we get 𝑔(𝑡)=𝑔(𝑥)+𝑔^′(𝑥)+)(𝑡−𝑥_𝑡𝑥(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢,𝑡∈[0,𝐴],𝐴>0, (42) And, by Lemmas 1, 2, and 3, we get 𝐿^∗_𝑚,𝑝𝑔;𝑥=𝑔(𝑥)+𝐿^∗_𝑚,𝑝^𝑡_𝑥(𝑡−𝑢)𝑔^′′(𝑢),𝐿𝑑𝑢,𝑥_∗𝑚,𝑝(𝑥≤𝐿𝑔;𝑥−𝑔)_{∗𝑚,𝑝𝑡𝑥}(𝑡−𝑢)𝑔^′′(𝑢)𝑑𝑢,𝑥≤𝐿^∗_𝑚,𝑝^𝑡_𝑥|(𝑡−𝑢)|𝑔^′′(𝑢)𝑑𝑢,𝑥≤𝐿^∗_𝑚,𝑝(𝑡−𝑥)²,𝑥𝑔^′′. (43) Using Lemma 5, we obtain 𝐿^∗_𝑚,𝑝𝑔;𝑥−𝑔(𝑥)≤𝑥²𝑚+𝑝_𝑞−[𝑚]_𝑞²−𝑚+𝑝_𝑞[𝑚]²_𝑞𝑔^′′+𝑥𝑚+𝑝_𝑞[𝑚]²_𝑞𝑔^′′. (44) On the other hand, by the definition of 𝐿^∗_𝑚,𝑝(𝑓;𝑞;𝑥), we have 𝐿^∗_𝑚,𝑝𝑓;𝑥≤𝑓. (45) Now 𝐿^∗_𝑚,𝑝(𝑥≤𝐿𝑓;𝑥−𝑓)_∗𝑚,𝑝−(𝑥+𝐿𝑓−𝑔;𝑥𝑓−𝑔)_∗𝑚,𝑝(𝑥≤𝑔;𝑥−𝑔)𝑓−𝑔+𝑥²𝑚+𝑝_𝑞−[𝑚]_𝑞²−𝑚+𝑝_𝑞[𝑚]²_𝑞𝑔^′′+𝑥𝑚+𝑝_𝑞[𝑚]²_𝑞𝑔^′′. (46) Hence taking infimum on the right hand side over all 𝑔∈𝑊², we get 𝐿^∗_𝑚,𝑝𝑓;𝑥−𝑓(𝑥)≤𝐶𝐾₂𝑓,𝛿²_𝑚(𝑥). (47) In view of the property of 𝐾-functional, we get 𝐿^∗_𝑚,𝑝𝑓;𝑥−𝑓(𝑥)≤𝐶𝑤₂𝑓,𝛿_𝑚(𝑥). (48)
This completes the proof of the theorem.

Let 𝐶^∗_𝐵[0,𝑝+1] be the space of all bounded functions for which lim_𝑥→∞𝑓(𝑥) is finite.

Theorem 9. Let 𝑓∈𝐶^∗_𝐵[0,𝑝+1] be such that 𝑓^′, 𝑓^′′∈𝐶^∗_𝐵[0,𝑝+1], and the sequence (𝑞_𝑚) satisfies (14). Then the following equality holds: lim_𝑚→∞[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝(𝑥)=𝑥()𝑓;𝑞;𝑥−𝑓1−𝑥2𝑓^′′ (49) uniformly on [0,𝑝+1].

Proof. By the Taylor’s formula we may write 𝑓(𝑡)=𝑓(𝑥)+𝑓^′(𝑥)+1)(𝑡−𝑥2𝑓^′′(𝑥)(𝑡−𝑥)²+𝑟(𝑡,𝑥)(𝑡−𝑥)², (50) where 𝑟(𝑡,𝑥) is the remainder term and lim_𝑡→𝑥𝑟(𝑡,𝑥)=0. Applying 𝐿^∗_𝑚,𝑝(𝑓;𝑞;𝑥) to (50), we obtain [𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝(𝑥=[𝑚]𝑓;𝑞;𝑥−𝑓)_𝑞𝑚𝐿^∗_𝑚,𝑝𝑡−𝑥;𝑞;𝑥𝑓^′(𝑥)+[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝(𝑡−𝑥)²𝑓;𝑞;𝑥^′′(𝑥)2+[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝𝑟(𝑡,𝑥)(𝑡−𝑥)²;𝑞;𝑥. (51) By the Cauchy-Schwartz inequality, we have 𝐿^∗_𝑚,𝑝𝑟(𝑡,𝑥)(𝑡−𝑥)²≤;𝑞;𝑥𝐿^∗_𝑚,𝑝𝑟²()×𝑡,𝑥;𝑞;𝑥𝐿^∗_𝑚,𝑝𝑟(𝑡,𝑥)(𝑡−𝑥)⁴;𝑞;𝑥. (52) Observing that 𝑟²(𝑥,𝑥)=0 and 𝑟²(⋅,𝑥)∈𝐶^∗_𝐵[0,𝑝+1], then it follows from Theorem 8 that 𝐿^∗_𝑚,𝑝𝑟²(𝑡,𝑥);𝑞;𝑥=𝑟²(𝑥,𝑥)=0, (53) uniformly with respect to 𝑥∈[0,𝑝+1]. Now from (52), (53), and Lemma 5, we get lim_𝑚→∞[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝𝑟(𝑡,𝑥)(𝑡−𝑥)²;𝑞;𝑥=0. (54) Finally using Lemmas 4 and 5, we get the following: lim_𝑚→∞[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝𝑓;𝑞;𝑥−𝑓(𝑥)=lim_𝑚→∞[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝𝑡−𝑥;𝑞;𝑥𝑓^′(𝑥)+lim_𝑚→∞[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝(𝑡−𝑥)²𝑓;𝑞;𝑥^′′(𝑥)2+lim_𝑚→∞[𝑚]_𝑞𝑚𝐿^∗_𝑚,𝑝𝑟(𝑡,𝑥)(𝑡−𝑥)²=𝑥();𝑞;𝑥1−𝑥2𝑓^′′(𝑥). (55)
This completes the proof of the theorem.

4. Example

Let us take 𝑓(𝑥)=1+sin(−6.5𝑥²). We compute error estimation [26] by using modulus of continuity for operators (5) to the function 𝑓(𝑥)=1+sin(−6.5𝑥²) shown in Table 1 with the help of MATLAB and Algorithm 1.

Table 1 

Error estimation table.

𝑠𝑦𝑚𝑠  𝑥; 𝑛= [20,30,50]; 𝑝=30;  𝑞= 0.9; 𝑑𝑎𝑡=𝑧𝑒𝑟𝑜𝑠(3,4); 𝑓𝑜𝑟   𝑖= 1 : 3    𝑚=𝑛(𝑖); 𝑒𝑟𝑟𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒=𝑖𝑛𝑙𝑖𝑛𝑒𝑐ℎ𝑎𝑟𝑥2(𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟(𝑚+𝑝,𝑞)−𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟(𝑚,𝑞))2−𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟(𝑚+𝑝,𝑞)(𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟(𝑚,𝑞))2       +𝑥𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑚+𝑝,𝑞(𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟(𝑚,𝑞))2×𝑎𝑏𝑠𝑑𝑖𝑓𝑓1+sin−6.5𝑥.2,2;   𝑑𝑎𝑡(𝑖,1)=𝑛(𝑖);   𝑑𝑎𝑡(𝑖,2)=𝑒𝑟𝑟𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒(0.2);   𝑑𝑎𝑡(𝑖,3)=𝑒𝑟𝑟𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒(0.5);   𝑑𝑎𝑡(𝑖,4)=𝑒𝑟𝑟𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒(0.8); end

Algorithm 1 

Error estimate algorithm.

For 𝑚=20, 30, and 50, the convergence of operators (5) to function 𝑓(𝑥)=1+sin−6.5𝑥² (56) is illustrated in Figure 1 with help of Algorithm 2.

𝑛= [20,30,50];𝑝=30;𝑞=0.9; 𝑓𝑜𝑟  𝑗= 1 : 3   𝑚=𝑛(𝑗);   𝑎=  [1 : 100];   𝑖=1;   𝑓𝑜𝑟  𝑥= 0 : 0.01 : 1    𝑡=0;    𝑓𝑜𝑟  𝑘= 0 :  m +𝑝     𝑧=1;     𝑓𝑜𝑟  𝑠= 0 :  m +𝑝−𝑘−1      𝑧=𝑧(1−𝑞𝑠𝑥);     𝑒𝑛𝑑     𝑡=𝑡+𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑓𝑎𝑐𝑡𝑚+𝑝,𝑞𝑥𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑓𝑎𝑐𝑡𝑘,𝑞𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑓𝑎𝑐𝑡𝑚+𝑝−𝑘,𝑞𝑘𝑧      ×1+sin−6.5𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑘,𝑞𝑞𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑚,𝑞2;     𝑒𝑛𝑑     𝑎(𝑖)=𝑡;     𝑖=𝑖+1;    𝑒𝑛𝑑    𝑥= 0 : 0.01 : 1;    𝑖𝑓(𝑗==1)     𝑐=𝑝𝑙𝑜𝑡(𝑥,𝑎)     𝑠𝑒𝑡(𝑐,“Color”, “blue”, “LineWidth”, 2)    𝑒𝑙𝑠𝑒  𝑖𝑓(𝑗==2)     𝑐=𝑝𝑙𝑜𝑡(𝑥,𝑎,“g”)     𝑠𝑒𝑡(𝑐,“Color” ,“green” ,“LineWidth”, 2)    𝑒𝑙𝑠𝑒     𝑐=𝑝𝑙𝑜𝑡(𝑥,𝑎,“r”)     𝑠𝑒𝑡(𝑐,“Color” ,“red” ,“LineWidth” ,2)    𝑒𝑛𝑑    ℎ𝑜𝑙𝑑  𝑜𝑛 𝑒𝑛𝑑 𝑥=  [0 : 0.01 : 1]; 𝑦=1+sin(−6.5𝑥2); 𝑎=𝑝𝑙𝑜𝑡(𝑥,𝑦,“−−k” )

Algorithm 2 

For the generalized 𝑞-Bernstein-Schurer operators.

Figure 1