Chapter 3

Stone Weierstrass Approximation The-orem3.1 The Bernstein Polynomials

These polynomials give an explicit description of a sequence of polynomials which con-verge uniformly to a continuous function. Recall that if you have a bounded functiondefined on some set S with values in Y.

∥ f∥∞≡ sup{∥ f (x)∥ : x ∈ S}

This is one way to measure distance between functions.

Lemma 3.1.1 The following estimate holds for x ∈ [0,1] and m≥ 2.

m

∑k=0

(mk

)(k−mx)2 xk (1− x)m−k ≤ 1

4m

Proof: First of all, from the binomial theorem

m

∑k=0

(mk

)(et(k−mx)

)xk (1− x)m−k = e−tmx

m

∑k=0

(mk

)(etk)

xk (1− x)m−k

= e−tmx (1− x+ xet)m ≡ e−tmxg(t)m , g(0) = 1,g′ (0) = g′′ (0) = x

Take a derivative with respect to t twice.

m

∑k=0

(mk

)(k−mx)2 et(k−mx)xk (1− x)m−k

= (mx)2 e−tmxg(t)m +2(−mx)e−tmxmg(t)m−1 g′ (t)

+e−tmx[m(m−1)g(t)m−2 g′ (t)2 +mg(t)m−1 g′′ (t)

]Now let t = 0 and note that the right side is m(x− x2)≤ m/4 for x ∈ [0,1] . Thus

m

∑k=0

(mk

)(k−mx)2 xk (1− x)m−k = mx−mx2 ≤ m/4 ■

With this preparation, here is the first version of the Weierstrass approximation theorem.I will allow f to have values in a complete normed linear space. Thus, f ∈ C ([0,1] ;X)where X is a Banach space, Definition 2.5.31. Thus this is a function which is continuouswith values in X as discussed earlier with metric spaces.

Theorem 3.1.2 Let f ∈C ([0,1] ;X) and let the norm be denoted by ∥·∥ .

pm (x)≡m

∑k=0

(mk

)xk (1− x)m−k f

(km

).

Then these polynomials converge uniformly to f on [0,1].

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