5.7. FUNCTIONS OF MANY VARIABLES 133

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 5.7.1 Let f be a function in C (R;X) for X a normed linear space whereR ≡∏

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.

Lemma 5.7.2 Let f ∈ C1 ([0,1]) and let pm (x) ≡ ∑mk=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

)

=m

∑k=1

m(m−1)!(m− k)!(k−1)!

xk−1 (1− x)m−k f(

km

)−

m−1

∑k=0

(mk

)xk (m− k)(1− x)m−1−k f

(km

)

=m−1

∑k=0

m(m−1)!(m−1− k)!k!

xk (1− x)m−1−k f(

k+1m

)−

m−1

∑k=0

m(m−1)!(m−1− k)!k!

xk (1− x)m−1−k f(

km

)

=m−1

∑k=0

m(m−1)!(m−1− k)!k!

xk (1− x)m−1−k(

f(

k+1m

)− f

(km

))