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B. Multivariate Bernstein Polynomials

Here we give the proofs of the theorems in
Section 7.4.

**Theorem B..1**
Let

be a continuous function. Then
the two-dimensional Bernstein polynomials

converge pointwise to

for

.

*Proof*.
Let

be a fixed point. Because of
Theorem

7.2 we have

for all

and

. The
second summand is smaller than

for

because

is the Bernstein polynomial for

, and the first summand is
smaller than

for

because

is the (one-dimensional) Bernstein polynomial
for

. Q.E.D.

**Definition B..2** (Multivariate Bernstein Polynomials)
Let

and

be a function of

variables.
The polynomials

are called the multivariate Bernstein polynomials of

.

**Theorem B..3** (Pointwise Convergence)
Let

be a continuous function. Then the
multivariate Bernstein polynomials

converge
pointwise to

for

.

*Proof*.
By applying Theorem

7.2 to each summand in

we see that given an

there are

,...,

such that

for all

. Q.E.D.

**Theorem B..5** (Uniform Convergence)
Let

be a continuous function. Then the
multivariate Bernstein polynomials

converge
uniformly to

for

.

*Proof*.
We first note that because of the uniform continuity of

on

we have

Given an

, we can find such a

. In order to
simplify notation we set

and

.

always lies in

. We have to estimate

and to that end we split the sum into two parts, namely

where

means summation over all

with

(where

) and

, and

where

means summation over the remaining terms.
For

we have

We will now estimate

. In the sum

the inequality

holds, i.e.,

Hence at least one of the summands on the left hand side is greater
equal

. Without loss of generality we can assume this
is the case for the first summand:

Thus, using Lemma

B.4,

We can now estimate

. Since

is continuous on a compact set

exists.

For

large enough we have

and thus

which completes the proof. Q.E.D.

A reformulation of this fact is the following corollary.

**Corollary B..6**
The set of all polynomials is dense in

.

**Theorem B..7** (Error Bound for Lipschitz Condition)
If

is a continuous function satisfying the
Lipschitz condition

on

, then the inequality

holds.

*Proof*.
Abbreviating notation we set

. We will use the Lipschitz condition,
Corollary

A.7, and
Lemma

B.4.

This completes the proof. Q.E.D.

**Theorem B..8** (Asymptotic Formula)
Let

be a

function and

, then

*Proof*.
We define the vector

through

, where the

are the integers over which we sum in

. Using
Theorem

A.14 we see

where

. Summing this equation like the sum
in

we obtain

since many terms vanish or can be summed because of
Lemma

B.4. Noting

we can
apply the same technique as in the proof of
Theorem

B.5 for estimating the last sum
in the last equation, i.e., splitting the sum into two parts for

and

. Hence we see that for all

this sum is less equal

for all sufficiently
large

, which yields the claim. Q.E.D.

Clemens Heitzinger
2003-05-08