New effective moduli of uniqueness and uniform a–priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory

Let U and V be complete separable metric spaces, Vu compact in V and G : U × V → IR a continuous function. For a large class of (usually non–constructive) proofs of uniqueness theorems ∧ u ∈ U, v1, v2 ∈ Vu ( G(u, v1) = inf v∈Vu G(u, v) = G(u, v2) → v1 = v2 ) one can extract an effective modulus of uniqueness Φ by logical analysis, i.e. ∧ u ∈ U, v1, v2 ∈ Vu, n ∈ IN ( 2 ∧ i=1 ( G(u, vi)− inf v∈Vu G(u, v) ≤ 2−Φun ) → dV (v1, v2) ≤ 2 −n ) . Since Φ does not depend on v1, v2 it is an a–priori estimate, which generalizes the notion of strong unicity in Chebycheff approximation theory. This applies to uniqueness proofs in Chebycheff approximation, ‖ · ‖1–approximation of f ∈ C[0, 1] and best uniform approximation by polynomials having bounded coefficients. Here we continue our proof–theoretic analysis started in [21]. A simplification of a proof by Young/Rice and a variant due to Borel are analysed yielding explicit moduli Φ and uniform a–priori (lower) estimates for strong unicity which are significantly better than those obtained from de La Vallée Poussin’s uniqueness proof in [21]. It is explained how the numerical content of the three proofs depends on the logical form whereby certain analytical lemmas (e.g. the alternation theorem) play a key–role. The numerical results obtained from Young/Rice’s proof are best and substantially improve known estimates obtained by D. Bridges by an n (=dimension of the Haar space) in the exponent.