# 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.

Lemma B..4   For all

For all we have and hence

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