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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