Faster approximation to multivariate functions by combined Bernstein-Taylor operators

Oktay Duman

doi:10.1515/dema-2025-0129

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Faster approximation to multivariate functions by combined Bernstein-Taylor operators

Oktay Duman

Veröffentlicht/Copyright: 26. Mai 2025

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Abstract

In this article, we incorporate multivariate Taylor polynomials into the definition of the Bernstein operators to get a faster approximation to multivariate functions by these combined operators. We also give various numerical simulations including graphical illustrations and error estimations. Our results improve not only the linear approximation by classical Bernstein polynomials but also the nonlinear approximation obtained by max-product operations.

Keywords: multivariate Bernstein polynomials; multivariate max-product Bernstein operators; multivariate Taylor oplynomials; rate of convergence; modulus of continuity

MSC 2010: 41A10; 41A25; 41A58

1 Introduction

The approximation of multivariate functions using Bernstein polynomials is a well-established topic. In this article, we focus on enhancing the rate of approximation by modifying Bernstein polynomials with the help of Taylor polynomials. The initial idea in this direction was introduced by Kirov [1] for the approximation of univariate functions. In this article, we first extend Kirov’s approach to the multivariate case, developing a framework to approximate multivariate functions more effectively. Second, we apply the same method to the nonlinear Bernstein operators modified using max-product operations. These modifications also lead to improved approximation results compared to the classical case. Our findings are supported by graphical representations and numerical values, which illustrate the superiority of the proposed modifications over the classical Bernstein operators.

In the classical approximation theory, one of the most effective known methods for (uniformly) approximating a multivariate function f that is continuous over the unit hypercube K is to use Bernstein polynomials B n ( f ) given by

(1.1) B n ( f ; x ) ≔ ∑ k = 0 n p n , k ( x ) f ( x k , n )

for n ∈ N and x = ( x 1 , … , x m ) ∈ K = [ 0 , 1 ] m , where x k , n ≔ k n and p n , k ( x ) ≔ ∏ i = 1 m n k i x i k i ( 1 − x i ) n − k i for k = ( k 1 , … , k m ) with k i ∈ { 0 , … , n } ( i = 1 , … , m ) . In (1.1), the symbol ∑ k = 0 n denotes the multi-summation ∑ k 1 = 0 n … ∑ k m = 0 n . There are many references in the literature on Bernstein polynomials and their variants [2–12]. Also, especially for the univariate version of (1.1), denoted by B n ( f ) , we recommend the pioneering book by Lorentz [13]. Let us note that multivariate Bernstein polynomials can also be defined on the simplex domain S = { x = ( x 1 , … , x m ) ∈ [ 0 , 1 ] m : x 1 + x 2 + … x m ≤ 1 } , which is a subset of the unit hypercube [ 0 , 1 ] m . However, to broaden the scope of approximation in this article, we considered the definition given in (1.1) instead.

It is well known that the rate of convergence for approximation to continuous functions by Bernstein polynomials is of order 1 n . For the univariate case, we know from Lorentz that (see page 22 of [13]), if f ′ belongs to the Lipschitz class Lip 1 , then ∣ B n ( f ; x ) − f ( x ) ∣ ≤ C n for some positive constant C . It is interesting to note that this last estimate for Bernstein polynomials cannot be improved by assuming the existence of the second or higher derivatives of f . The main objective of this article is to increase this rate of convergence by modifying (1.1). For this purpose, we will incorporate multivariate Taylor polynomials into (1.1) and show that the approximation of functions by these combined operators is faster than Bernstein polynomials.

As usual, for a given r ∈ N 0 ≔ N ∪ { 0 } , C r ( K ) denotes the family of functions f : K → R having continuous r th order partial derivatives. We also consider the next notation:

∣ u ∣ = u 1 + … + u m , u ! = u 1 ! … u m ! , x u = x 1 u 1 … x m u m , ∂ u f = ∂ 1 u 1 … ∂ m u m = ∂ ∣ u ∣ f ∂ x 1 u 1 … ∂ x m u m , ∣ u ∣ ≤ r .

With this terminology, the combined Bernstein-Taylor operators are defined by

(1.2) B n [ r ] ( f ; x ) ≔ ∑ k = 0 n p n , k ( x ) ∑ ∣ u ∣ ≤ r ∂ u f ( x k , n ) u ! ( x − x k , n ) u

for f ∈ C r ( K ) . It is easy to check that if r = 0 , then (1.2) reduces to (1.1), that is, B n [ 0 ] = B n . It should be noted that the univariate version of (1.2) was first examined by Kirov [1]. However, to the best of our knowledge, the multidimensional version of such a work has not been studied yet. The difficulty is probably due to the complexity of higher order partial derivatives of functions of several variables. In this article, we will overcome this difficulty by exploiting the properties of multivariate Taylor polynomials and operators (1.2). We will then show that the newly constructed combined operators exhibit superior approximation properties compared to the classical Bernstein polynomials (see Theorem 2.1 and Remark 2.1 in the next section).

Another important aspect of this work is that we will also obtain similar results for nonlinear Bernstein operators. Recall that if we replace the sum operation ∑ in (1.1) by the max operation ∨ , then we get the max-product (nonlinear) Bernstein operators given by

(1.3) B n [ M ] ( f ; x ) ≔ ∨ k = 0 n p n , k ( x ) ⋅ f ( x k , n ) ∨ k = 0 n p n , k ( x )

for x ∈ [ 0 , 1 ] m , where ∨ k = 0 n ≔ ∨ k 1 = 0 n … ∨ k m = 0 n . The univariate version of (1.3), denoted by B n [ M ] , was studied by Bede et al. [7], while the bivariate case was examined in [8]. Then, we know from [7,8] that the rate of convergence to positive and continuous functions by these max-product operators is again of order 1 n and, for arbitrary f , this order of approximation cannot be improved. Also, for many subclasses of functions, one can find the order 1 n , which is better than the order of approximation by the linear Bernstein operators (see, for instance, [7,8]). We should note here that the positivity condition of functions is due to the following fact, which is essential in the approximation:

f ( x ) = ∨ k = 0 n p n , k ( x ) ⋅ f ( x ) ∨ k = 0 n p n , k ( x )

holds for f ( x ) ≥ 0 since p n , k ( x ) ≥ 0 for all x ∈ [ 0 , 1 ] m . Now let C * r ( K ) denote the class of the functions f : K → R + = [ 0 , ∞ ) having continuous r th-order partial derivatives. Note that for functions of this class, even though f is itself positive, its partial derivatives do not have to be positive on K . Then, by using again multivariate Taylor polynomials, we modify the max-product Bernstein operators as follows:

(1.4) B n [ r , M ] ( f ; x ) ≔ ∨ k = 0 n p n , k ( x ) ⋅ ∑ ∣ u ∣ ≤ r ∂ u f ( x k , n ) u ! ( x − x k , n ) u ∨ k = 0 n p n , k ( x )

for f ∈ C * r ( K ) . In this case, we will show that these combined operators can also achieve faster approximation than the operators B n [ M ] .

2 Approximation by combined Bernstein-Taylor operators in (1.2) and (1.4)

Let m ∈ N be given and consider the moment function defined by

φ x ( t ) ≔ ∣ t − x ∣ m = ( t 1 − x 1 ) 2 + ( t 2 − x 2 ) 2 + … + ( t m − x m ) 2

for t = ( t 1 , t 2 , … , t m ) ∈ K . Then, we first need the next lemma.

Lemma 2.1

For α > 0 , x ∈ K and n ∈ N , we have

B n [ 0 ] ( φ x α ; x ) = B n ( φ x α ; x ) ≤ D n α ⁄ 2 ,

where the positive constant D depends only on m and α .

Proof

We observe from (1.1) that

B n ( φ x α ; x ) = ∑ k = 0 n p n , k ( x ) ∣ x − x k , n ∣ m α ≤ ∑ k = 0 n p n , k ( x ) ∑ i = 1 m x i − k i n α ≤ m α ∑ k = 0 n p n , k ( x ) ∑ i = 1 m x i − k i n α = m α ∑ i = 1 m B n ( φ x i α ; x i ) ,

where B n denotes the univariate Bernstein polynomial as stated earlier. On the other hand, we may write from page 15 of [13] that

B n ( φ x i α ; x i ) ≤ A α n α ⁄ 2 ( i = 1 , … , m )

holds for a constant A α depending only on α . Now by combining these results and letting D ≔ m α + 1 A α , the proof is completed.□

In our recent paper [14], using the concept of modulus of continuity ω ( h , δ ) , δ > 0 , of a continuous function h on K defined by

ω ( h , δ ) ≔ sup { ∣ h ( t ) − h ( x ) ∣ : ∣ t − x ∣ m < δ , x , t ∈ K } ,

we have also proved the next lemma.

Lemma 2.2

[14] For m , r ∈ N , b , x ∈ K , δ > 0 and f ∈ C r ( K ) , we have

f ( x ) − ∑ ∣ u ∣ ≤ r ∂ u f ( b ) u ! ( x − b ) u ≤ M ∣ x − b ∣ m r 1 + ∣ x − b ∣ m δ ∨ ∣ u ∣ = r ω ( ∂ u f , δ ) ,

for some M > 0 depending only upon m and r.

Now using the supremum norm ∥ ⋅ ∥ on K , we present our first approximation theorem.

Theorem 2.1

Let m ∈ N and r ∈ N 0 be given. Then, for every n ∈ N and f ∈ C r ( K ) , we have

(2.1) ∥ B n [ r ] ( f ) − f ∥ ≤ C ∨ ∣ u ∣ = r ω ∂ u f , 1 n ( r + 1 ) ⁄ 2 ,

where the positive constant C depends only upon m and r. Then, for every f ∈ C r ( K ) , (2.1) implies the uniform convergence of B n [ r ] ( f ; x ) to f on K .

Proof

We first observe that the result is clear for r = 0 . Now let r ∈ N . For a given x ∈ K and a function f ∈ C r ( K ) , we may write from (1.2) that

∣ B n [ r ] ( f ; x ) − f ( x ) ∣ ≤ ∑ k = 0 n p n , k ( x ) f ( x ) − ∑ ∣ u ∣ ≤ r ∂ u f ( x k , n ) u ! ( x − x k , n ) u

holds. Letting b = x k , n in Lemma 2.2, we obtain the inequality

∣ B n [ r ] ( f ; x ) − f ( x ) ∣ ≤ M ( ∨ ∣ u ∣ = r ω ( ∂ u f , δ ) ) ∑ k = 0 n p n , k ( x ) ∣ x − x k , n ∣ m r 1 + ∣ x − x k , n ∣ m δ

for some M > 0 . Then, we obtain

∣ B n [ r ] ( f ; x ) − f ( x ) ∣ ≤ M ( ∨ ∣ u ∣ = r ω ( ∂ u f , δ ) ) B n ( φ x r ; x ) + 1 δ B n ( φ x r + 1 ; x ) .

Now Lemma 2.1 implies that

∣ B n [ r ] ( f ; x ) − f ( x ) ∣ ≤ D ⋅ M ( ∨ ∣ u ∣ = r ω ( ∂ u f , δ ) ) 1 n r ⁄ 2 + 1 δ n ( r + 1 ) ⁄ 2

holds for some constant D . Combining the aforementioned estimates and taking δ ≔ δ n = 1 n ( r + 1 ) ⁄ 2 and C ≔ 2 D M the proof follows.□

Remark 2.1

From Theorem 2.1 if ∂ u f is Lipschitz continuous, then we obtain the next result:

∥ B n [ r ] ( f ) − f ∥ = O 1 n ( r + 1 ) ⁄ 2 .

This clearly shows that approximation to functions by our combined operators B n [ r ] is faster than the classical Bernstein polynomials B n . In the next section, we will present various numerical simulations that support this point.

Remark 2.2

By a simple translation in Theorem 2.1, we can also take any hyperrectangle H ≔ [ a 1 , b 1 ] × … × [ a m , b m ] instead of K = [ 0 , 1 ] m . Then, considering the nodes

x k , n H = a 1 + ( b 1 − a 1 ) k 1 n , … , a m + ( b m − a m ) k m n ,

the corresponding combined Bersntein-Taylor operators become

B n [ r ] , H ( f ; x ) ≔ ∑ k = 0 n p n , k H ( x ) ∑ ∣ u ∣ ≤ r ∂ u f ( x k , n H ) u ! ( x − x k , n H ) u ,

where

p n , k H ( x ) ≔ ∏ i = 1 m 1 ( a i − b i ) n n k i x i k i ( b i − x i ) n − k i

for k = ( k 1 , … , k m ) with k i ∈ { 0 , … , n } ( i = 1 , … , m ) .

Now, we focus on the approximation by the combined operators B n [ r , M ] in (1.4). We first need the next lemma.

Lemma 2.3

For α > 0 , x ∈ K , and n ∈ N , we have

B n [ 0 , M ] ( φ x α ; x ) = B n [ M ] ( φ x α ; x ) ≤ D n α ⁄ 2 ,

where the positive constant D depends only upon m and α .

Proof

As in the proof of Lemma 2.1, we may write that

B n [ M ] ( φ x α ; x ) ≤ m α ∑ i = 1 m B n [ M ] ( φ x i α ; x i ) ,

where B n [ M ] denotes the univariate max-product Bernstein operator as stated before. We know from [7] that the inequality

B n [ M ] ( φ x i ; x i ) ≤ 6 ( n + 1 ) 1 ⁄ 2

is satisfied for each i = 1 , … , m . Now, following the same steps as in [7], but just substituting φ x i α for φ x i , we arrive directly at the following inequality:

B n [ M ] ( φ x i α ; x i ) ≤ 6 α n α ⁄ 2 ( i = 1 , … , m ) .

Now combining the above results and also taking D ≔ m α + 1 6 α , the proof is completed.□

Then, we obtain the following result.

Theorem 2.2

Let m ∈ N and r ∈ N 0 . For every n ∈ N and f ∈ C * r ( K ) , we obtain

∥ B n [ r , M ] ( f ) − f ∥ ≤ C ∨ ∣ u ∣ = r ω ∂ u f , 1 n ( r + 1 ) ⁄ 2 ,

for some C > 0 . Furthermore, if ∂ u f ( ∣ u ∣ = r ) is Lipschitz continuous on K , we obtain

∥ B n [ r , M ] ( f ) − f ∥ = O 1 n ( r + 1 ) ⁄ 2 .

Proof

The case of r = 0 is clear. Let r ∈ N , x ∈ K and f ∈ C r ( K ) be given. Then, by using the fundamental properties of max-product operations [8] and also considering the similar steps as in the proof of Theorem 2.1, Lemma 2.2 implies, for every δ > 0 , that the inequality

∣ B n [ r , M ] ( f ; x ) − f ( x ) ∣ ≤ M ( ∨ ∣ u ∣ = r ω ( ∂ u f , δ ) ) B n [ M ] ( φ x r ; x ) + 1 δ B n ( φ x r + 1 ; x ) .

holds for some M > 0 . Now, from Lemma 2.3, we obtain

∣ B n [ r , M ] ( f ; x ) − f ( x ) ∣ ≤ D ⋅ M ( ∨ ∣ u ∣ = r ω ( ∂ u f , δ ) ) 1 n r ⁄ 2 + 1 δ n ( r + 1 ) ⁄ 2

for some positive constant D . Therefore, taking δ ≔ δ n = 1 n ( r + 1 ) ⁄ 2 and C ≔ 2 D M , the first part of the proof is completed. The remaining part of the proof is already straightforward.□

3 Numerical simulations

In this section, we give some numerical simulations of Theorems 2.1 and 2.2. Now let m = 2 . Then, as a target function, we first consider the next function:

(3.1) f ( x ) ≔ sin 25 x − 1 2 2 + 25 y − 1 2 2

for x = ( x , y ) ∈ K. In Figure 1, we show both the 3D graph of this function and its level curves. Note that larger values are shown in lighter color. Since f ∈ C 2 ( K ) , Theorem 2.1 implies that B n [ r ] ( f ) is uniformly convergent f on K for each r ∈ N 0 , which is illustrated in Figure 2 for r = 0 , 1, 2 and n = 5 , 10, 15, respectively. In Figure 3, we also show the level curves of this approximation for the same values of r and n .

Figure 1

Graph of the target function f given by (3.1): (a) 3D plot and (b) level curves.

$Figure 2 Approximation to f f by B n [ r ] ( f ) {{\mathbb{B}}}_{n}^{\left[r]}(f) : (a) r = 0 r=0 and n = 5 n=5 , 10, 15, respectively, (b) r = 1 r=1 and n = 5 n=5 , 10, 15, respectively, and (c) r = 2 r=2 and n = 5 n=5 , 10, 15, respectively.$

Figure 2

Approximation to f by B n [ r ] ( f ) : (a) r = 0 and n = 5 , 10, 15, respectively, (b) r = 1 and n = 5 , 10, 15, respectively, and (c) r = 2 and n = 5 , 10, 15, respectively.

$Figure 3 Level curves in the approximation to f f by B n [ r ] ( f ) {{\mathbb{B}}}_{n}^{\left[r]}(f) : (a) r = 0 r=0 and n = 5 n=5 , 10, 15 respectively, (b) r = 1 r=1 and n = 5 n=5 , 10, 15, respectively, and (c) r = 2 r=2 and n = 5 n=5 , 10, 15, respectively.$

Figure 3

Level curves in the approximation to f by B n [ r ] ( f ) : (a) r = 0 and n = 5 , 10, 15 respectively, (b) r = 1 and n = 5 , 10, 15, respectively, and (c) r = 2 and n = 5 , 10, 15, respectively.

Now it is also possible to calculate numerically the errors in this approximation. By using the absolute error e j [ r ] ≔ ∣ B n [ r ] ( f ) − f ∣ at the n e = 30 × 30 regular grid points of K , we can evaluate the following pointwise errors:

e mean [ r ] ≔ 1 n e ∑ j = 1 n e e j [ r ] (the mean error), e m s [ r ] ≔ 1 n e ∑ j = 1 n e e j [ r ] 2 (the mean square error)

for the values of r = 0 , 2 and n = 20 , 40, 80, respectively. Then, we can check from Table 1 that approximation by the combined operators is better than the classical Bernstein polynomials.

Table 1

Pointwise errors in the approximation

(a) The mean errors
n	e mean [ 0 ]	e mean [ 2 ]
20	0.4210173	0.3787726
40	0.2767969	0.2054019
80	0.1609748	0.0779949

(b) The mean sequare errors
n	e m s [ 0 ]	e m s [ 2 ]
20	0.4749271	0.4220906
40	0.3152598	0.2344957
80	0.1855133	0.0918319

Finally, to observe the approximation in Theorem 2.2, we choose the following target function g , which is indicated in Figure 4:

(3.2) g ( x ) ≔ 1 + sin ( 10 x y )

for x = ( x , y ) ∈ K = [ 0 , 1 ] 2 . Since g ∈ C * 2 ( K ) , Theorem 2.2 gives that B n [ r , M ] ( g ) is uniformly convergent to g on K . This is illustrated in Figures 5 and 6 for r = 0 , 1, 2 and n = 5 , 10, 15, respectively.

Figure 4

Graph of the target function g given by (3.2): (a) 3D plot and (b) level curves.

$Figure 5 Approximation to g g by B n [ r , M ] ( g ) {{\mathbb{B}}}_{n}^{\left[r,M]}\left(g) : (a) r = 0 r=0 and n = 5 n=5 , 10, 15, respectively, (b) r = 1 r=1 and n = 5 n=5 , 10, 15, respectively, and (c) r = 2 r=2 and n = 5 n=5 , 10, 15, respectively.$

Figure 5

Approximation to g by B n [ r , M ] ( g ) : (a) r = 0 and n = 5 , 10, 15, respectively, (b) r = 1 and n = 5 , 10, 15, respectively, and (c) r = 2 and n = 5 , 10, 15, respectively.

$Figure 6 Level curves in the approximation to g g by B n [ r , M ] ( g ) {{\mathbb{B}}}_{n}^{\left[r,M]}\left(g) : (a) r = 0 r=0 and n = 5 n=5 , 10, 15, respectively, (b) r = 1 r=1 and n = 5 n=5 , 10, 15, respectively, and (c) r = 2 r=2 and n = 5 n=5 , 10, 15, respectively.$

Figure 6

Level curves in the approximation to g by B n [ r , M ] ( g ) : (a) r = 0 and n = 5 , 10, 15, respectively, (b) r = 1 and n = 5 , 10, 15, respectively, and (c) r = 2 and n = 5 , 10, 15, respectively.

4 Concluding remarks

By using multivariate Taylor polynomials, we have modified the classical Bernstein polynomial and its nonlinear version based on max-product operations. These modifications enable us to obtain more efficient approximation results than (linear and nonlinear) Bernstein operators.

It is important to note that while this modification significantly improves the approximation speed, it also introduces additional computational complexity depending on the choice of the Taylor polynomial with degree r . Our results indicate that even for small values of r , a substantial improvement in convergence can be achieved with a reasonable computational cost. However, as r increases, the additional computational burden may become more significant. A detailed analysis of this trade-off, particularly regarding the practical implementation of the proposed approach, is an important topic for future research.

We should also note that, so far, such modifications have been focused more on Shepard operators by using not only Taylor polynomials but also Bernoulli, Lidstone, and Hermite polynomials [14–18]. Therefore, it will be interesting to develop Bernstein operators with these polynomials in future research. In addition, further investigations into optimizing the balance between computational complexity and convergence rate could enhance the practical applicability of the proposed method.

okitayduman@gmail.com

Acknowledgements

The author would like to express his gratitude to the reviewers for their insightful comments and suggestions, which have improved the clarity and impact of this work.

Funding information: Author states no funding involved.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: Prof. Oktay Duman is a member of the Editorial Board of the journal Demonstratio Mathematica, but was not involved in the review process of this article.
Ethical approval: The conducted research is not related to either human or animals use.
Data availability statement: Not applicable.

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Received: 2024-12-10

Revised: 2025-03-03

Accepted: 2025-04-15

Published Online: 2025-05-26

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multivariate Bernstein polynomials; multivariate max-product Bernstein operators; multivariate Taylor oplynomials; rate of convergence; modulus of continuity

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