Effective Moduli from Ineffective Uniqueness Proofs. An Unwinding of de La Vallée Poussin's Proof for Chebycheff Approximation

We consider uniqueness theorems in classical analysis having the form (+) ∀u ∈ U, v1, v2 ∈ Vu ( G(u, v1) = 0 = G(u, v2) → v1 = v2 ) , where U, V are complete separable metric spaces, Vu is compact in V and G : U × V → IR is a constructive function. If (+) is proved by arithmetical means from analytical assumptions (++) ∀x ∈ X∃y ∈ Yx∀z ∈ Z ( F (x, y, z) = 0 ) only (where X, Y, Z are complete separable metric spaces, Yx ⊂ Y is compact and F : X × Y × Z → IR constructive), then we can extract from the proof of (++) → (+) an effective modulus of uniqueness, i.e. (+ + +) ∀u ∈ U, v1, v2 ∈ Vu, k ∈ IN ( |G(u, v1)|, |G(u, v2)| ≤ 2 −Φuk → dV (v1, v2) ≤ 2 −k ) . Such a modulus Φ can e.g. be used to give a finite algorithm which computes the (uniquely determined) zero of G(u, ·) on Vu with prescribed precision if it exists classically. The extraction of Φ uses a proof–theoretic combination of functional interpretation and pointwise majorization. If the proof of (++) → (+) uses only simple instances of induction, then Φ is a simple mathematical operation (on a convenient standard representation of X, e.g. on f together with a modulus of uniform continuity for X = C[0, 1]). Various uniqueness theorems in best approximation theory have the form (+) and are proved using only analytical tools of the form (++). We analyse the most common proof of uniqueness for the best Chebycheff approximation of f ∈ C[0, 1] by polynomials of degree ≤ n given by de La Vallée Poussin and obtain explicit moduli of uniqueness and uniform constants of strong unicity. In a subsequent paper two further proofs of this uniqueness will be analysed yielding better estimates (due to the fact that mainly (++)–lemmas are used) which allow us to improve results obtained prior by D. Bridges significantly.