# 7.4 Multivariate Bernstein Polynomials

In order to keep the formulae simple we will again only consider functions defined on the multidimensional intervals , i.e., the unit cube in  . Using affine transformations it is straightforward to apply the formulae and results to arbitrary intervals. The proofs from this section can be found in Appendix B.

To illustrate the general concept we first look at the two-dimensional case. We obtain the desired approximation by first approximating one variable and then the second.

Theorem 7..6   Let be a continuous function. Then the two-dimensional Bernstein polynomials

converge pointwise to for .

We define now the multivariate Bernstein polynomials as follows.

Definition 7..7 (Multivariate Bernstein Polynomials)   Let and be a function of variables. The polynomials

are called the multivariate Bernstein polynomials of .

We note that is a linear operator.

Theorem 7..8 (Pointwise Convergence)   Let be a continuous function. Then the multivariate Bernstein polynomials converge pointwise to  for .

But using this straightforward method we can only prove pointwise convergence.

Lemma 7..9   For all

For all we have and hence

Theorem 7..10 (Uniform Convergence)   Let be a continuous function. Then the multivariate Bernstein polynomials converge uniformly to  for .

A reformulation of this fact is the following corollary. It ensures that all functions considered in TCAD applications can be approximated arbitrarily well.

Corollary 7..11   The set of all polynomials is dense in .

By presupposing more knowledge about the rate of change of the function, namely a Lipschitz condition, an error bound is obtained.

Theorem 7..12 (Error Bound for Lipschitz Condition)   If is a continuous function satisfying the Lipschitz condition

on , then the inequality

holds.

The following asymptotic formula gives us information about the rate of convergence.

Theorem 7..13 (Asymptotic Formula)   Let be a function and , then

The asymptotic formula states that the rate of convergence depends only on the partial derivatives . This is noteworthy, since it is often the case that the smoother a function is and the more is known about its higher derivatives, the more properties can be proven, but in this case only the second order derivatives play a role.

Clemens Heitzinger 2003-05-08