444 CHAPTER 16. APPROXIMATION OF FUNCTIONS AND THE INTEGRAL

which implies maxy∈H maxx∈K h(x,y)≥max(x,y)∈K×H h(x,y) . The other inequality is alsoobtained.

Let f ∈C (Rp;X) where Rp = [0,1]p . Then let x̂p ≡ (x1, ...,xp−1) . By Theorem 16.1.2,if n is large enough,

maxxp∈[0,1]

∥∥∥∥∥ n

∑k=0

f

(·, k

n

)(nk

)xk

p (1− xp)n−k−f (·,xp)

∥∥∥∥∥C([0,1]p−1;X)

<ε

2

Now f(·, k

n

)∈C (Rp−1;X) and so by induction, there is a polynomial pk (x̂p) such that

maxx̂p∈Rp−1

∥∥∥∥pk (x̂p)−(

nk

)f

(x̂p,

kn

)∥∥∥∥X<

ε

(n+1)2

Thus, letting p(x)≡ ∑nk=0pk (x̂p)xk

p (1− xp)n−k ,

∥p−f∥C(Rp;X) ≤ maxxp∈[0,1]

maxx̂p∈Rp−1

∥∥p(x̂p,xp)−f (x̂p,xp)∥∥

X < ε

where p is a polynomial with coefficients in X .In general, if Rp ≡∏

pk=1 [ak,bk] , note that there is a linear function lk : [0,1]→ [ak,bk]

which is one to one and onto. Thus l (x)≡ (l1 (x1) , ..., lp (xp)) is a one to one and onto mapfrom [0,1]p to Rp and the above result can be applied to f ◦ l to obtain a polynomial p with∥p−f ◦ l∥C([0,1]p;X) < ε. Thus

∥∥p◦ l−1−f∥∥

C(Rp;X) < ε and p◦ l−1 is a polynomial. Thisproves the following theorem.

Theorem 16.2.1 Let f be a function in C (R;X) for X a normed linear space where R ≡∏

pk=1 [ak,bk] . Then for any ε > 0 there exists a polynomial p having coefficients in X such

that ∥p−f∥C(R;X) < ε .

These Bernstein polynomials are very remarkable approximations. It turns out that if fis C1 ([0,1] ;X) , then limn→∞ p′n (x)→ f ′ (x) uniformly on [0,1] . This all works for func-tions of many variables as well, but here I will only show it for functions of one variable. Iassume the reader knows about the derivative of a function of one variable.

Lemma 16.2.2 Let f ∈C1 ([0,1]) and let

pm (x)≡m

∑k=0

(mk

)xk (1− x)m−k f

(km

)be the mth Bernstein polynomial. Then in addition to ∥pm− f∥[0,1]→ 0, it also follows that∥∥p′m− f ′

∥∥[0,1]→ 0

Proof: From simple computations,

p′m (x) =m

∑k=1

(mk

)kxk−1 (1− x)m−k f

(km

)

−m−1

∑k=0

(mk

)xk (m− k)(1− x)m−1−k f

(km

)