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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