Bernstein-Type Operators on the Unit Disk

We construct and study sequences of linear operators of Bernstein-type acting on bivariate functions defined on the unit disk. To this end, we study Bernstein-type operators under a domain transformation, we analyze the bivariate Bernstein–Stancu operators, and we introduce Bernstein-type operators on disk quadrants by means of continuously differentiable transformations of the function. We state convergence results for continuous functions and we estimate the rate of convergence. Finally some interesting numerical examples are given, comparing approximations using the shifted Bernstein–Stancu and the Bernstein-type operator on disk quadrants.


Preliminaries
In 1912, S. Bernstein ([2]) published a constructive proof of the Weierstrass approximation theorem that affirms that every continuous function f (x) defined on a closed interval can be uniformly approximated by polynomials. For a given function f ∈ C[0, 1], Bernstein constructed a sequence of polynomials (lately called Bernstein polynomials) in the form for 0 x 1, and n 0.
Clearly, B n f is a polynomial in the variable x of degree less than or equal to n, and (1.1) can be seen as a linear operator that transforms functions defined on [0, 1] to polynomials of degree at most n.
Hence, in the sequel, we will refer to B n as the nth classical univariate Bernstein operator.
If we define Among others, classical Bernstein operators satisfy the following properties ( [13]): • They are linear and positive operators acting on the function f and preserve the constant functions as well as polynomials of degree 1, that is, The Bernstein operators admit a complete system of polynomial eigenfunctions. However, each eigenfunction depends on n and, thus, is associated with the nth Bernstein operator B n . Another inconvenience of Bernstein operators associated with an adequate function f is its slow rate of convergence toward f . For years, several modifications and extensions of Bernstein operators have been studied. The modifications have been introduced in several directions, and we only recall a few interesting cases and cite some papers. For instance, it is possible to substitute the values of the function on equally spaced points by other mean values such as integrals, as was stated in the pioneering papers of Durrmeyer ([9]) and Derriennic ([6, 7]). In [4], the operator is modified in order to preserve some properties of the original function. Another group of modifications given by the transformation of the function by means a convenient continuous and differentiable functions is analyzed in [5]; and, of course, the extension of the Bernstein operators to the multivariate case. The most common extension of the Bernstein operator is defined on the unit simplex in higher dimensions ( [1,13,16,17], among others), since the basic polynomials (1.2) can be easily extended to the simplex.
In this paper, we are interested in finding an extension of the Bernstein operator to approximate functions defined on the unit disk. In this way, we will consider two kinds of modifications: by transformation of the argument of the function to be approximated, and by definition of an adequate basis of functions as (1.2). We present and study two Bernstein-type approximants, and we compare them by means of several examples.
The structure of the paper is as follows. Section 2 is devoted to collecting the properties of univariate Bernstein-type operators that we will need along the paper. In Sect. 3, we recall the method introduced by Stancu ( [17]) for obtaining Bernstein-type operators in two variables by the successive application of Bernstein operators in one variable. In Sect. 5 and Sect. 6, we define the shifted nth Bernstein-Stancu operator and the shifted nth Bernstein-type operator and study their respective approximation properties. Section 7 is devoted to describing an extension of certain linear combinations of univariate Bernstein operators that give a better order of approximation. The last section is devoted to analyzing several examples, comparing the approximation results for both Bernstein-type operators on the disk, and the linear combinations introduced in Sect. 7.

Univariate Bernstein-Type Operators
In this section, we recall the modified univariate Bernstein-type operators that we will need later. We start by shifting the univariate Bernstein operator.
Using the change of variable  α β we have that Bernstein basis on [α, β] (see Fig. 1) satisfies the following properties: • p n,k (α) = δ 0,k and p n,k (β) = δ k,n , where, as usual, δ ν,η denotes the Kronecker delta, • If n = 0, then p n,k (x; [α, β]) has a unique local maximum on [α, β] at x = (β − α) k n + α. This maximum takes the value For every function f defined on I = [α, β], we can define the shifted univariate nth Bernstein operator as Note that B n [ f (x), I ] is a polynomial of degree at most n. In this way, In the sequel, we will use the following Bernstein-type operator studied in [5] and [10]: where τ is any function continuously differentiable as many times as necessary, such that τ (0) = 0, τ(1) = 1, and τ (x) > 0 for x ∈ [0, 1]. Throughout this work, it will be sufficient for τ to be continuously differentiable.
In [5], the following identities were given: We have the following result. Since as n → +∞, the result follows from taking the limit on both sides of C τ We also introduce the following shifted Bernstein-type operator: where τ (x) is any function that is continuously differentiable, such that τ (α) = α, τ (β) = β, and τ (x) > 0 for x ∈ [α, β]. Since as n → +∞, the result follows from taking the limit on both sides of C τ

Bivariate Bernstein-Stancu operators
In 1963, Stancu [17] studied a method for deducing polynomials of Bernstein type of two variables. This method is based on obtaining an operator in two variables from the successive application of Bernstein operators of one variable. Let φ 1 ≡ φ 1 (x) and φ 2 ≡ φ 2 (x) be two continuous functions such that φ 1 < φ 2 on [0, 1]. Let ⊆ R 2 be the domain bounded by the curves y = φ 1 (x), y = φ 2 (x), and the straight lines x = 0, x = 1. For every function f (x, y) defined on , taking into view (3.1) let us define the function where 0 t 1.
The nth Bernstein-Stancu operator is defined as where each n k is a nonnegative integer associated with the kth node x k = k/n, and t is given by (3.1). Writing (3.3) explicitly, we have If we denote by B (t) n the univariate Bernstein operator acting on the variable t, then the Bernstein-Stancu operator can be written as We have the following representation of B n in terms of a matrix determinant.

Remark 3.2
Observe that the step size of the partition of the x-axis is 1/n and, for a fixed node x k = k/n, the step size of the partition of the t-axis is 1/n k . Therefore, the step size of the partition of the y-axis is 1/m k , where and, thus, We point out that, in general, B n [ f (x, y), ] is not a polynomial. However, it is possible to obtain polynomials by an appropriate choice of φ 1 , φ 2 , and n k . For instance: (1) The Bernstein-Stancu operator on the unit square Q = [0, 1] × [0, 1] (see for instance [13], [17]) is obtained by letting φ 1 (x) = 0 and φ 2 (x) = 1. Hence, for a function f defined on Q, we get n k j=0 f k n , j n k p n,k (x) p n k , j (y).
Note that when n k is independent of k (e.g., n k = m for some positive integer m), B n is the tensor product of univariate Bernstein operators on Q.

Theorem 3.3 ([17]) Let f be a continuous function on
converges uniformly to f (x, y) as n → +∞.
Stancu only gave a detailed proof of the approximation properties of B n on triangles. In Sect. 5, we consider a slightly general operator and prove the uniform convergence on any bounded domain , and we recover Stancu's result when = T 2 .
is a Bernstein operator on Q. Indeed, for every function f defined on Q, we define the function F : Q → R as Then, using the transformation x = 2u − 1 and y = 2v − 1 which maps Q into Q, we get (2) An alternative way to obtain the Bernstein-Stancu operator on the simplex T 2 is by considering the Duffy transformation which maps Q into T 2 . Let f be a function defined on T 2 . We can define the function Then, the operator is a Bernstein-type operator on the simplex since, using the Duffy transformation, we get is not a polynomial unless n − k − n k 0. We recover the usual Bernstein-Stancu operator on the simplex by setting n k = n − k.
(3) Consider the unit ball in R 2 : For every function f defined on B 2 , we can define the function F : Q → R 2 as The operator is a Bernstein operator on the unit ball since Observe that, in this case, In contrast with the previous two cases, there is no obvious choice of n k such that is a polynomial. Nevertheless, notice that for y = 0, we have and for x = 0 we have is a polynomial on the x-and y-axes for any choice of n k . In Fig. 2, the representation of the mesh in this case for n = n k = 20 is given. (4) Let which maps each quadrant to Q. The corresponding Bernstein operators on the quadrants are: Indeed, for every function f defined on B 2 , we can define the functions on Q: Then, In this case, observe that for k = 0, the mesh corresponding to B 1 and B 2 , and similarly to B 3 and B 4 , coincides on the y-axis (see Fig. 3). Moreover, for j = 0, the mesh corresponding to adjacent quadrants coincides on the x-axis. Therefore, we can define a piecewise Bernstein operator on B 2 as follows: (4.1)

Proposition 4.1 For any function f on
Similarly, for y = 0 and is continuous on the x-and y-axes.

Shifted Bernstein-Stancu Operators
Motivated by the examples of Bernstein operators on different domains introduced in the previous section, now we define the shifted nth Bernstein-Stancu operator and study its approximation properties. Let φ 1 and φ 2 be two continuous functions, and let I = [a, b] be an interval such that φ 1 < φ 2 on I . Let ⊂ R 2 be the domain bounded by the curves y = φ 1 (x), y = φ 2 (x), and the straight lines x = a, x = b. Observe that for a fixed x ∈ I , the polynomials p n,k (y; For every function f (x, y) defined on , define the function The shifted nth Bernstein-Stancu operator is defined as , where n k = n − k or n k = k for all 0 k n. Written in terms of the univariate Bernstein basis, we get The following result plays an important role when studying the convergence of the shifted Bernstein-Stancu operator.
where B n denotes the univariate shifted Bernstein operator acting on the variable x. Since B n converges uniformly for a continuous function, we have n .
(v) Finally, if f (x, y) = y 2 in (5.1), then we get Then, Observe that Together with (5.2), we get If n k = n − k, then n k=0 p n,k (x; , I , , I .
The convergence of the operator is clear from Lemma 5.1 and Volkov's theorem ( [18]). Now, we study the approximation properties of the shifted Bernstein-Stancu operators.

Theorem 5.3 Let f be a continuous function on . Then,
Proof Let δ 1 , δ 2 > 0 be real numbers.
Note that on we have B n [1, ] = 1, Taking into account the inequality (see, for instance, [16,17]) Therefore,  Recall that the univariate shifted Bernstein satisfies the following Voronowskaya type asymptotic formula: Let f (x) be bounded on the interval I , and let x 0 ∈ I at which f (x 0 ) exists. Then, Now, we give an analogous result for the Bernstein-Stancu operator.

Theorem 5.4 Let f (x, y) be a bounded function on
y φ 2 (x)}, and let (x 0 , y 0 ) ∈ be a point at which f (x, y) admits second-order partial derivatives, and φ i (x 0 ), i = 1, 2, exist. Then, Proof Let us write the Taylor expansion of f (u, v) at the point (x 0 , y 0 ): Applying B n to both sides, we get: where we have omitted for brevity. We deal with each term separately. From Lemma 5.1 (ii), we get B n [u − x 0 ] u=x 0 = 0. Next, from the proof of Lemma 5.1 (iii), we have But using (5.3), we get Similarly,

Now we deal with the last term
Fix a real number ε > 0. Then, there is a real number δ > 0 such that if ||(u, v) − (x 0 , y 0 )|| < δ, then |h(u, v)| < ε. Let S δ be the set of k and j such that 1 δ 2 F k n , j n k > 1. Then, Moreover, we have Thus, Putting all the above together, we get and the result follows.

Shifted Bernstein-Type Operators
We define the shifted bivariate Bernstein-type operator. Let φ 1 and φ 2 be two continuous functions, and let I = [a, b] be an interval such that φ 1 < φ 2 on I . Let ⊂ R 2 be the domain bounded by the curves y = φ 1 (x), y = φ 2 (x), and the straight lines where τ is any continuously differentiable function on I , such that τ (a) = a, τ(b) = b, and τ (x) > 0 for x ∈ I , and for each fixed x ∈ I , σ x is any continuously For every function f (x, y) defined on , define the function for 0 u 1, and 0 v 1, where φ i , i = 1, 2, are defined in (5.1).

The shifted bivariate Bernstein-type operator is defined as
for (x, y) ∈ , where n k = n − k or n k = k for 0 k n. Written in terms of the univariate classical Bernstein basis, we get  f (x, y). Now, we study shifted Bernstein-type operators defined on each quadrant of B 2 , denoted by B i for i = 1, 2, 3, 4. We will choose T and n k such that, for any function, the approximation given by Bernstein-type operators on each quadrant is a polynomial. (i) For x ∈ [0, 1], let τ (x) = x 2 and, for each fixed value of x, let σ x (y) =

The polynomials L [2k]
n f (x) satisfy the recurrence relation and, if f (2k) exists at a point x ∈ [0, 1], then Using (7.1), we can obtain the following explicit expressions for the constants α j 's, In [14], May considers a slightly more general operator is a polynomial of degree 2 k n. However, May proved that if f (2 k+1 ) exists, then and Although we do not study the approximation behavior of these operators here, the numerical experiments in the following section suggest a better rate of convergence than B n and C n .

Numerical Experiments
In this section, we present numerical experiments where we compare the shifted Bernstein-Stancu operator B n on B 2 , and the shifted Bernstein-type operator C n in (6.1). To do this, we consider different functions defined on B 2 . For each function f (x, y), we compute B n [ f (x, y), We use a set of points randomly distributed on the unit disk (generated by mesh function in Mathematica) to compare the function to its approximations. For B n [ f (x, y), B 2 ], we use 630 points (x i , y i ). We set z i = f (x i , y i ), 1 i 630, andẑ i equal to the value of B n [ f (x, y), B 2 ] at the respective point (x i , y i ), and compute the root-mean-square error (RMSE) as follows: Similarly, for C n [ f (x, y), B 2 ], we use randomly distributed 1082 points (x j ,ȳ j ). We set w j = f (x j ,ȳ j ), 1 j 1082, andw j equal to the value of C n [ f (x, y), B 2 ] at the respective point (x j ,ȳ j ), and compute the RMSE as follows: .
In each case, we plot the RMSE for increasing values of n using Mathematica. For each operator, the set of points used to compute the RMSE consists of a fixed number of points. On the other hand, the number of mesh points used to represent each operator depends on n.
We represent C n [ f (x, y), B 2 ] on each quadrant using different colors as shown in Fig. 4. We take n = 100, then the mesh required to obtain the operator for each quadrant consists of 20200 points.
For B n [ f (x, y), B 2 ], we take n = 200. Then, the mesh required to obtain the operator for all the unit disk consists of 40401 points.
We note that the operator C n requires two evaluations at the mesh points on the common boundaries of two adjacent quadrants. Therefore, the operator B n needs a smaller number of evaluations than the operator C n since, for a fixed n, B n and C n are composed of (n + 1) 2 and 2 (n + 1) (n + 2) evaluations, respectively.
Additionally, we compute the RMSE for S [1] n [ f (x, y), 2 j ] and R [1] n [ f (x, y), 2 j ] using the same set of randomly distributed points as before.

Example 1
First, we consider the continuous function The graph of f (x, y) is shown in Fig. 5, and the approximations C n [ f (x, y), and B n [ f (x, y), B 2 ] are shown in Fig. 6. We list the RMSE of both approximations for different values of n in Table 1 and plot them together in Fig. 7, where the characteristic slow convergence inherited from the univariate Bernstein operators is observed. Moreover, the corresponding RMSEs are shown in Table 2 and Fig. 8 for S [1] n [ f (x, y), 2 j ] and R [1] n [ f (x, y), 2 j ], where a seemingly better approximation behavior can be observed.  f (x, y).

Example 2
Now, we consider the continuous periodic function Its graph is shown in Fig. 9. It can be observed in Fig. 10 that the approximation error for both operators is larger at the maximum and minimum values of the function. Table   Fig. 11 contain further evidence of this larger error. Moreover, in comparison with the previous example, it seems that the rate convergence of C n [g(x, y), B 2 ] is significantly faster than the rate of convergence of B n [g(x, y), B 2 ]. Table 4 and Fig. 12 show the errors corresponding to R [1] n [ f (x, y), 2 j ] and S [1] n [ f (x, y), 2 j ]. In comparison with B n and C n , R [1] n , and S [1] n appear to have a better approximation behavior.

Example 3
Here, we consider the continuous function h(x, y) = e x 2 −y 2 − x y, (x, y) ∈ B 2 , (see Fig. 13). Both approximations are shown in Fig. 14, and their respective RMSEs are listed in Table 5 and plotted in Fig. 15. Observe that, in this case, the RMSEs for both approximations are significantly smaller than in the previous examples. Moreover, based on Fig. 15, it seems that for sufficiently large values of n, the rate of convergence of both approximations is considerably similar to each other. Table 6 and Fig. 16 also show similar approximation behavior between S [1] n [ f (x, y), 2 j ] and R [1] n [ f (x, y), 2 j ].

Example 4
In this numerical example, we are interested in observing the behavior of shifted Bernstein-type and shifted Bernstein-Stancu operators at jump discontinuities. Let us consider the following discontinuous function: if 0.5 x 2 + y 2 0.8, 0.5, if 0.8 < x 2 + y 2 1.
The graph of η(x, y) is shown in Fig. 17 and the approximations are shown in Fig. 14. It is interesting to observe the behavior of the approximations at the points of jump   discontinuities and, thus, we have included Fig. 19, where we show a cross-sectional view of the approximations with increasing values of n. As in the univariate case, it seems that the Gibbs phenomenon does not occur. Finally, Table 7 and Fig. 20 expose a significantly slow convergence rate for this discontinuous function in comparison with the previous continuous examples. As can be seen in Table 8 and Fig. 21, it seems   [1] n [η(x, y); 2 j ] S [1] n [η(x, y); 2 j ] that a better approximation can be obtained with the operators S [1] n [ f (x, y), 2 j ] and R [1] n [ f (x, y), 2 j ].