Chapter 9

Weierstrass Approximation Theorem9.1 The Bernstein Polynomials

This short chapter is on the important Weierstrass approximation theorem. It is about ap-proximating an arbitrary continuous function uniformly by a polynomial. It will be as-sumed only that f has values in C and that all scalars are in C. First here is some notation.

Definition 9.1.1 α = (α1, · · · ,αn) for α1 · · ·αn positive integers is called a multi-index.For α a multi-index, |α| ≡ α1 + · · ·+αn and if x ∈ Rn,

x =(x1, · · · ,xn) ,

and f a function, definexα ≡ xα1

1 xα22 · · ·x

αnn .

A polynomial in n variables of degree m is a function of the form

p(x) = ∑|α|≤m

aα xα .

Here α is a multi-index as just described. You could have aα have values in a normedlinear space.

The following estimate will be the basis for the Weierstrass approximation theorem. Itis actually a statement about the variance of a binomial random variable.

Lemma 9.1.2 The following estimate holds for x ∈ [0,1].

m

∑k=0

(mk

)(k−mx)2 xk (1− x)m−k ≤ 1

4m

Proof: By the Binomial theorem,

m

∑k=0

(mk

)(etx)k(1− x)m−k =

(1− x+ etx

)m. (9.1.1)

Differentiating both sides with respect to t and then evaluating at t = 0 yields

m

∑k=0

(mk

)kxk (1− x)m−k = mx.

Now doing two derivatives of 9.1.1 with respect to t yields

∑mk=0(m

k

)k2 (etx)k (1− x)m−k = m(m−1)(1− x+ etx)m−2 e2tx2

+m(1− x+ etx)m−1 xet .

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