Proof Mining in Analysis: Computability and Complexity

Proof Mining consists in extracting from prima facie non-constructive proofs constructive information. For that purpose many techniques mostly based on negative translation, realizability, functional interpretation, A-translation and combinations thereof have been developed. After introducing various different approaches to Proof Mining we present an application of ‘monotone functional interpretation’ to a modification of a classical proof from 1921 of uniqueness of L1-approximation of continuous functions by polynomials of degree ≤ n. From that proof we extracted the first effective uniform rate of strong uniqueness (also called ‘uniform modulus of uniqueness’) for L1-approximation, which existence had been studied for 80 years but always in a non-constructive manner. For that reason, previous results only stated the dependencies of the modulus but explicit constants had never been obtained. We also show how this modulus of uniqueness can be used to compute the actual best L1-approximation of the given continuous functions on a compact interval. Finally, we analyze the complexity of the resulting algorithm using the tools of Computable Analysis (cf. [Ko91] and [Wei00]). The first part of this work (the extraction of the uniform modulus of uniqueness) has been concluded, the last two parts are under development. This report also includes indications of future areas of work.