# Iterated Bernstein polynomial approximations

###### Abstract

Iterated Bernstein polynomial approximations of degree for continuous function which also use the values of the function at , , are proposed.
The rate of convergence of the classic Bernstein polynomial approximations is significantly improved
by the iterated Bernstein polynomial approximations without increasing the degree of the polynomials.
The close form expression of the limiting iterated Bernstein polynomial approximation of degree
when the number of the iterations approaches infinity is obtained.
The same idea applies to the -Bernstein polynomials and the Szasz-Mirakyan approximation.
The application to numerical integral approximations which gives surprisingly good results is also discussed.

MSC: 41A10; 41A17; 41A25.

Keywords: Bernstein polynomials; Bézier curves;
Convexity preservation; Iterated Bernstein polynomials; Numerical
integration; -Bernstein polynomials; Rate of approximation; the
Szasz-Mirakyan operators.

## 1 Introduction

The Bernstein polynomials [1] have been used for approximations of functions in many areas of mathematics and other fields such as smoothing in statistics and constructing Bézier curves [see 2, 3, for examples] which have important applications in computer graphics. One of the advantages of the Bernstein polynomial approximation of a continuous function is that it approximates on uniformly using only the values of at , . In case when the evaluation of is difficult and expensive, the Bernstein polynomial approximation is preferred.

The properties of the Bernstein polynomial approximation have been studied extensively by many authors for decades. However the slow optimal rate of convergence of the classical Bernstein polynomial approximation makes it not so attractive. Many authors have made tremendous efforts to improve the performance of the classical Bernstein polynomial approximation. Among many others, Butzer[4] introduces linear combinations of the Bernstein polynomials and Phillips[5] proposes the -Bernstein polynomials which is a generalization of the classical Bernstein polynomial approximation. However, Butzer[4]’s approximation involves not only the Bernstein polynomials of degree but also degree of which requires more sampled values of the function to be approximated at the rather than uniform partition points of . The -Bernstein polynomial approximates a function only when . For , it seems that has to be an analytic complex function on disk , , so that the -Bernstein polynomial approximation of degree has a better rate of convergence, , than the best rate of convergence, , of the classical Bernstein polynomial approximation of degree [see 6, 7, for example]. If , the -Bernstein polynomial approximation of degree uses the sampled values of the function at nonuniform partition points of . These points except are attracted toward when is getting larger so that the approximation becomes worse in the the neighborhood of the right end-point. This is a serious drawback of the -Bernstein polynomial approximation which limits the scope of its applications.

In this paper, we propose a simple procedure to generalize and improve the classical Bernstein polynomial approximation by repeatedly approximating the errors using the Bernstein polynomial approximations. This method involves only the iterates of the Bernstein operator applied on the base Bernstein polynomials of degree and the sampled values of the function being approximated at the same set of uniform partition points of . The improvement made by the -Bernstein polynomial approximation with properly chosen can be achieved by the iterated Bernstein polynomials without messing up the right boundary.

## 2 Preliminary Results About the Classical Bernstein Polynomial

Let be a function on . The classical Bernstein polynomial of degree is defined as

(1) |

where is called the Bernstein operator and , , are called the Bernstein basis polynomials. Note that the Bernstein polynomial of degree , , uses only the sampled values of at , . Note also that for ,

is the density function of beta distribution . Let be a binomial random variable. Then , , , and . The error of is

(2) |

Let be a member of , the set of all continuous functions that have continuous first derivatives. . Let the modulus of continuity of the th derivative be

About the rate of convergence of we have the following well known results [see 8].

###### Theorem 1.

Suppose , . For each

where is a constant depending on only. One can choose and .

The result according to is due to Popoviciu[9]. The order of approximation of by arbitrary polynomials is given by the theorem of Dunham Jackson [10]

###### Theorem 2 (Dunham Jackson).

Suppose , . For each there exists a polynomial of degree at most so that

where is a constant depending on only. If , one can choose .

The following is a result of Voronovskaya [11] about the asymptotic formula of the Bernstein polynomial approximation.

###### Theorem 3 (E. Voronovskaya).

Suppose that has second derivative . Then

(3) |

where is a sequence of functions which converge to 0 as .

From Theorem 3 it follows that the best rate of convergence of , as , is even if has continuous second or higher order derivatives [8]. This is not as good as in the case of arbitrary polynomial approximation in which if has continuous th derivative then the rate of convergence of a sequence of arbitrary polynomials of degree at most can be at least [10]. Bernstein [12] generalizes this asymptotic formula to contain terms up to the th derivative and proposes a polynomial constructed based on both and , . Butzer [4] considers some combinations of Bernstein polynomials of different degrees and shows that they have better rate of convergence which is much faster than . Costabile et al [13] generalize the linear combinations of the Bernstein polynomials proposed by of [4], [14] and [15]. The -Bernstein polynomials of [5] has better rate of convergence. However, if , the -Bernstein polynomials of function do not approximate . For , the -Bernstein polynomials do approximate at a rate of but has to be analytic in a complex disk with radius greater than . The analyticity of may be too restrictive for applications. Even if we are sure that is analytic, we have to deal with the choice of . In some cases, the approximations are very sensitive to the choice of .

## 3 The Iterated Bernstein Polynomials and the Rate of Convergence

The error is also a continuous function on whose values at , , depend on , , only. So we can approximate this error function by the Bernstein polynomial and then subtract the approximated error function from to obtain the second order Bernstein polynomial of degree

(4) |

This idea is closely related to, although was not initiated by, the proposal of Bernstein [12] in which the second derivative rather than the error of the Bernstein polynomial is approximated. Inductively,

(5) |

This iteration procedure can be performed further until a satisfactory approximation precision is achieved because the error can be estimated by .

###### Lemma 4.

Generally the -th order Bernstein polynomial of degree can be written as

(6) |

Define . Then the error of the -th Bernstein polynomial of degree can be written as

(7) |

where is the identity operator.

###### Proof.

The limit of , as , has been given by Kelisky and Rivlin [16]. A short and elementary proof of [16]’s result is given by [17]. After we finished the first version of this paper, we realized that [18] obtained the formula (7) and investigated the properties of using simulation method. The cost of is only some simple algebraic calculations in addition to the evaluation of at , .

About the iterates of the Bernstein operator we have the following result.

###### Lemma 5.

For ,

(9) |

where , and

(10) |

When ,

(11) |

###### Proof.

The theorem can be easily proved by induction and the fact that the Bernstein operator is linear. ∎

###### Theorem 6.

The -th Bernstein polynomial approximation can be calculated inductively as

(12) |

Clearly, for every , preserves linear functions. Therefore

(13) |

Expression (12) can easily implemented in computer languages using iterative algorithm. Define indicator functions

(14) |

Then , and, by Theorem 6, (12) and (13) can be simplified as

(15) | ||||

(16) |

(17) |

The following theorem shows that the iterated Bernstein polynomials , like the classical ones, have no error at the endpoints of .

###### Theorem 7.

For any function defined on and any integer ,

(18) |

###### Proof.

Clearly, for each , can be written as

where is an row vector, and

If ,

Define square matrix where

That is

It is easy to see that is nonsingular and have all the eigenvalues in among them exactly two are ones which correspond to eigenvectors and . We have the following theorem.

###### Theorem 8.

For ,

(19) |

where , the st order unit matrix. If ,

(20) |

More importantly, we have

###### Theorem 9.

The “optimal” Bernstein polynomial approximation of degree is

(21) |

where

(22) |

Moreover, preserves linear functions.

###### Proof.

Since all the eigenvalues of matrix are in and exactly two of them are ones, all the eigenvalues of matrix are in and exactly two of them are zeros. Thus , the zero matrix. Because preserves linear functions for any positive integer , so does . This can also be proved by the following facts that

and that is true provided that is linear.

∎

Numerical examples (see §6) show that the maximum absolute approximation error seems to be minimized by “optimal” Bernstein polynomial approximation if is infinitely differentiable. For nonsmooth functions such as and fixed , it seems that the maximum absolute approximation error is minimized by the iterated Bernstein polynomial approximation for some .

The next theorem shows that if then is indeed a better polynomial approximation of than the classical Bernstein polynomial.

###### Theorem 10.

Suppose that , and . Then

(23) |

where is a constant depending on and only.

###### Remark 3.1.

From this theorem with and , we see that if has continuous second derivative then the rate of convergence of the second Bernstein polynomial approximation is at least .

###### Remark 3.2.

From Theorem 10 with we see that if has continuous fourth derivative, then the rate of convergence of can be as fast as . This seems the fastest rate that can reach even if has continuous fifth or higher derivatives.

###### Remark 3.3.

It can also be proved that if has continuous th derivative, then the rate of convergence of can be as fast as . Although these improvements upon are still not as good as those stated in Theorem 2, they are good enough for application in computer graphics and statistics.

###### Remark 3.4.

It is a very interesting project to investigate the relationship between and , and the rate of convergence of which is conjectured to be exponential.

## 4 The Derivatives and Integrals of and Applications

### 4.1 The Derivatives of

###### Theorem 11.

For any positive integers and ,

(24) |

where is the th forward difference operator with increment , ,

###### Proof.

If , it is well known that for any function

(25) |

Assume that (24) with is true for the th iterated Bernstein polynomial of any function . By (5) we have

(26) |

It follows from (25) and (9) that

(27) |

Combining (25), (9), (4.1), and (4.1) we arrive at

(28) |

The proof of (24) with and is complete by induction. Similarly (24) with and can be proved using induction. ∎

It is not hard to prove by adopting the method of [8] that

###### Theorem 12.

(i) If has continuous th derivative on , then for each fixed , as , converge to uniformly on .

(ii) If in bounded on and its th derivative exists at , then for each fixed , as , converge to .

Numerical examples show that the larger the is, the slower the above convergence is.

For any positive integers , the second derivative of the iterated Bernstein polynomial is

(29) |

It is well known that if is convex on , then and thus is also convex and on . So the classical Bernstein polynomials preserve the convexity of the original function and has nonnegative errors. However examples of §6 show that when the iterated Bernstein polynomial does not preserve the convexity of the original function unconditionally. The iterated Bernstein polynomials still preserve the monotonicity of if it is not too “flat” anywhere.

###### Theorem 13.

If is strictly increasing (decreasing) on , for any , is also strictly increasing (decreasing) on .

###### Proof.

The theorem is true for even if is increasing (decreasing), but not strictly, on . It suffices to prove the theorem when is strictly increasing on . Assume that the theorem is true for some . Since is strictly increasing on , are also strictly increasing on for all . ∎

###### Remark 4.1.

If , the condition of strict monotonicity is not necessary. However, if , the condition of strict monotonicity can be relaxed. For example, , if , , if , and , if . It can be shown that for in a neighborhood of .

### 4.2 The Integrals of

The following theorem is very useful for implementing the iterative algorithm in computer languages.

###### Theorem 14.

Suppose is continuous on . For and , we have

(30) |

where and

###### Corollary 15.

If is continuous on , , then for ,

(31) |

where is calculated based on .

###### Remark 4.2.

###### Theorem 16.

## 5 Iterated Szasz Approximation and Iterated -Bernstein Polynomial

The idea used to construct the iterated Bernstein polynomial approximation is simple and very effective. The same idea seems also applicable to other operators or approximations such as the Szasz operator [19] [or the Szasz-Mirakyan (Mirakja) operator] and the -Bernstein polynomial with . We will give some numerical examples in §6 and the analogues of results of Section 3 could be be obtained by using the analogue results about the rate of convergence of the Szasz-Mirakyan approximation [20]. We hope these would inspire more investigations with rigorous mathematics.

### 5.1 Iterated Szasz Approximation

The so-called Szasz-Mirakyan approximation is defined as

(33) |

where is defined on and . Note that, for , is the probability that where is the Poisson random variable with mean . Since the binomial probability can be approximated by for large , the Szasz-Mirakyan approximation can be viewed as an extension of the Bernstein polynomial approximation. The error of as an approximation of is

(34) |

Applying the Szasz-Mirakyan operator to , we have

(35) |

So we can define the second Szasz-Mirakyan approximation as

(36) |

###### Theorem 17.

(37) |

Clearly, for every , preserves linear functions and therefore

(38) |

Figure 4 gives an example of the iterated Szasz approximations.

### 5.2 Iterated -Bernstein Polynomial

Let be a real number. For any , define the -number

If is integer, then is called a -integer. For , the -binomial coefficient (Gaussian binomial) is defined by

So

where empty product is defined to be 1. Thus the ordinary binomial coefficient