Chapter 16

Approximation of Functions and the In-tegral

These topics are not about linear algebra, but linear algebra is used in a very essentialmanner so these topics are applications of Linear algebra to analysis. Many more examplescould be included but this book is long enough.

This chapter is just what the title indicates. It will involve approximating functions anda simple definition of the integral. This definition is sufficient to consider all piecewisecontinuous functions and it does not depend on Riemann sums. Thus it is closer to whatwas done in the 1700’s than in the 1800’s. However, it is based on the Weierstrass approx-imation theorem so it is definitely dependent on material which originated in the 1800’s.After this, is a very interesting application of ideas from linear algebra to prove Müntz’stheorems.

The notation C ([0,b] ;X) will denote the functions which are continuous with values in[0,b] with values in X which will always be a normed vector space. It could be C or R forexample.

16.1 Weierstrass Approximation TheoremAn arbitrary continuous function defined on an interval can be approximated uniformly bya polynomial, there exists a similar theorem which is just a generalization of this which willhold for continuous functions defined on a box or more generally a closed and bounded set.However, we will settle for the case of a box first. The proof is based on the followinglemma.

Lemma 16.1.1 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

)(tx)k (1− x)m−k = (1− x+ tx)m

Take a derivative and then let t = 1.

m

∑k=0

(mk

)k (tx)k−1 x(1− x)m−k = mx(tx− x+1)m−1

m

∑k=0

(mk

)k (x)k (1− x)m−k = mx

Then also,m

∑k=0

(mk

)k (tx)k (1− x)m−k = mxt (tx− x+1)m−1

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