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.
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