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Let U and V be complete separable metric spaces, Vu compact in V and G : U IR a continuous function. For a large class of (usually non--constructive) proofs of uniqueness theorems Vu G(u, v1 ) = inf G(u, v) = G(u, v2) v1 = v2 one can extract an e#ective modulus of uniqueness # by logical analysis, i.e.

In this paper we present another case study in the general project of proof mining which means the logical analysis of prima facie non-effective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation (developed in [17]) to analyze Cheney's simplification [6] of Jackson's original proof [10] from 1921 of the uniqueness of the best L 1 -approximation of continuous functions f # C[0, 1] by polynomials p # Pn of degree # n. Cheney's proof is non-effective in the sense that it is based on classical logic and on the non-computational principle WKL (binary Konig lemma). The result of our analysis provides the first e#ective (in all parameters f, n and #) uniform modulus of uniqueness (a concept which generalizes `strong uniqueness' studied extensively in approximation theory). Moreover, the extracted modulus has the optimal #-dependency as follows from Kroo [21]. The paper also describes how the uniform modulus of uniqueness can be used to compute the best L 1 -approximations of a fixed f # C[0, 1] with arbitrary precision. We use this result to give a complexity upper bound on the computation of the best L 1 -approximation in [24].

Abstract not found

A historical account is given of the development of methods for solving approximation problems set in normed linear spaces. Approximation of both real functions and real data is considered, with particular reference to L p (or l p ) and Chebyshev norms. As well as coverage of methods for the usual linear problems, an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters, and special attention is paid to some particular nonlinear approximating families. 1 Introduction The purpose of this paper is to give a historical account of the development of numerical methods for a range of problems in best approximation, that is problems which involve the minimization of a norm. A treatment is given of approximation of both real functions and data. For the approximation of functions, the emphasis is on the use of the Chebyshev norm, while for data approximation, we consider a wider range of criteria, including the other l ...

Abstract not found

We give an upper bound on the complexity of the sequence (p n ) n#N of best L 1 - approximations of a polynomial-time computable function on the interval [0, 1] from the space of polynomials of bounded degree Pn . The analysis makes essential use of the modulus of uniqueness for L 1 -approximation presented in [KO01]. 1

We report on the extraction (presented in [KO01]) of the first e#ective uniform modulus of uniqueness (this notion is defined in section 2) for best polynomial L 1-approximation of continuous functions from the ine#ective proof of uniqueness due to Cheney. The