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

Abstract

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 theoremsone can extract an effective modulus of uniqueness Φ by logical analysis, i.e.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.

Gödel's Functional Interpretation and Its Use in Current Mathematics

“…This, in particular, is the case for best approximation theory, where based on functional interpretation new results on the best Chebycheff as well as L 1 -approximation of functions in C[0, 1] by polynomials p ∈ P n of degree ≤ n (for the case of Chebycheff approximation also for more general so-called Haar spaces instead of polynomials) have been obtained ( [56,57,74]). Whereas the difficult cases of Chebycheff and L 1 -approximation deal with the special spaces C[0, 1] and P n the following much simpler uniqueness result applies to a general class of spaces.…”

“…This approach, also called 'proof mining', has led to a number of new effective quantitative results but also to new qualitative results on the independence of solutions from certain parameters (uniformity results). The following papers in analysis use directly such techniques or use results obtained by these techniques or have been guided by general logical metatheorems which were established using functional interpretations: [4,10,11,12,13,14,25,57,61,62,63,65,70,71,72,73,74,88]. For surveys see [67,68] and -though covering only results up to 2002 - [74] which explains in detail general aspects of applying functional interpretation to analysis.…”

The Use of a Logical Principle of Uniform Boundedness in Analysis

“…The predicate 'uniform' here means that the bound Φ does not depend on v ≤ ρ tuk. In [7] we have shown, how sentences (1) naturally arise in analysis and why such uniform bounds are of numerical interest (see also [5], [6]). Proofs in analysis can be formalized in a suitable base theory T ω plus certain (in general nonconstructive) analytical principles Γ (typically not derivable in T ω ).…”

“…Moreover we gave a perspicuous method for the the extraction of bounds from proofs using WKL and arbitrary axioms (4) by a new combination of functional interpretation with majorization which, in [7], was simplified even further to the monotone functional interpretation. In [5], [6] this was applied to concrete proofs in best approximation theory yielding new numerical estimates which improved known estimates significantly (see [7] for a discussion of these results). In [5] we also gave a detailed representation of IR, C[0, 1] and more general complete separable metric spaces and showed that e.g.…”

A Complexity Analysis of Functional Interpretations

“…This gives functional interpretations the ability of extracting programs and other effective data 4 under certain conditions from ineffective proofs (proof mining). Monotone-functional-interpretationbased proof mining has already produced important results in computational analysis and has helped to obtain new results in mathematical analysis (see, e.g., [26,28,29,31,32,35,36,37,38]). …”