6. An Application to Approximation 25

(6.2) Proposition. Suppose that T is a finite set. Let us fix an ≥ 0. Suppose

that for any m + 1 points τ1,... , τm+1 from T one can construct a function fx(τ)

(with x = (ξ1, . . . , ξm) depending on τ1,... , τm+1) such that

|g(τ) − fx(τ)| ≤ for τ = τ1,... , τm+1.

Then there exists a function fx(τ) such that

|g(τ) − fx(τ)| ≤ for all τ ∈ T.

Proof. For a τ ∈ T let us define a set A(τ) ⊂

Rm:

A(τ) = (ξ1, . . . , ξm) : g(τ) − fx(τ) ≤ .

In other words, A(τ) is the set of functions fx that approximate g within at the

point τ. Now A(τ) are convex sets (see Problem 1 below), and

A(τ1) ∩ . . . ∩ A(τm+1) = ∅

for all possible choices of m + 1 points τ1,... , τm+1 in T . Since T is finite, Helly’s

Theorem (Theorem 4.2) implies that the intersection of all sets A(τ) : τ ∈ T is

non-empty. It follows then that for a point

x ∈

τ∈T

A(τ)

we have |g(τ) − fx(τ)| ≤ for all τ ∈ T .

PROBLEMS.

1◦.

Let A(τ) be as in the proof of Proposition 6.2. Prove that A(τ) ⊂

Rm+1

is

a closed convex set.

2◦.

Show that for m ≥ 2 the set A(τ) is not compact.

To prove a version of Proposition 6.2 for infinite sets T , we must assume some

regularity of functions f1,... , fm.

(6.3) Proposition. Suppose there is a finite set of points σ1,... , σn in T such

that whenever fx = ξ1f1 + . . . + ξmfm and fx(σ1) = . . . = fx(σn) = 0, then ξ1 =

. . . = ξm = 0. Suppose further that for any set of m + 1 points τ1,... , τm+1 in T

one can construct a function fx (with x = (ξ1, . . . , ξm) depending on τ1,... , τm+1)

such that

|g(τ) − fx(τ)| ≤ for τ = τ1,... , τm+1.

Then there exists a function

fx(τ) such that

|g(τ) − fx(τ)| ≤ for all τ ∈ T.