118 CHAPTER 6. CONTINUOUS FUNCTIONS

6.10 Weierstrass ApproximationIt turns out that if f is a continuous real valued function defined on an interval, [a,b] thenthere exists a sequence of polynomials, {pn} such that the sequence converges uniformlyto f on [a,b]. I will first show this is true for the interval [0,1] and then verify it is true onany closed and bounded interval. First here is a little lemma which is interesting for its ownsake in probability. It is actually an estimate for the variance of a binomial distribution.

Lemma 6.10.1 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: Here are some observations. ∑mk=0(m

k

)kxk (1− x)m−k =

mxm

∑k=1

(m−1)!(k−1)!((m−1)− (k−1))!

xk−1 (1− x)(m−1)−(k−1)

= mxm−1

∑k=0

(m−1

k

)xk (1− x)m−1−k = mx

m

∑k=0

(mk

)k (k−1)xk (1− x)m−k

= m(m−1)x2m

∑k=2

(m−2)!(k−2)!(m−2− (k−2))!

xk−2 (1− x)(m−2)−(k−2)

= m(m−1)x2m−2

∑k=0

(m−2

k

)xk (1− x)(m−2)−k = m(m−1)x2

Now (k−mx)2 = k2−2kmx+m2x2 = k (k−1)+ k (1−2mx)+m2x2. From the above andthe binomial theorem, ∑

mk=0(m

k

)(k−mx)2 xk (1− x)m−k =

m

∑k=0

(mk

)k (k−1)xk (1− x)m−k +(1−2mx)

m

∑k=0

(mk

)kxk (1− x)m−k

+m2x2m

∑k=0

(mk

)xk (1− x)m−k = m(m−1)x2 +(1−2mx)mx+m2x2

= mx(1− x)≤ m14

Now let f be a continuous function defined on [0,1] . Let pn be the polynomial definedby

pn (x)≡n

∑k=0

(nk

)f(

kn

)xk (1− x)n−k . (6.5)

Theorem 6.10.2 The sequence of polynomials in 6.5 converges uniformly to f on[0,1]. These polynomials are called the Bernstein polynomials.