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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