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Proof mining in L_1approximation
, 2001
"... In this paper we present another case study in the general project of proof mining which means the logical analysis of prima facie noneffective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation (developed in [17]) to a ..."
Abstract

Cited by 10 (4 self)
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In this paper we present another case study in the general project of proof mining which means the logical analysis of prima facie noneffective 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 noneffective in the sense that it is based on classical logic and on the noncomputational 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].
On the Computational Complexity of Best ...Approximation
 In preparation
"... We give an upper bound on the complexity of the sequence (p n ) n#N of best L 1  approximations of a polynomialtime 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 p ..."
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We give an upper bound on the complexity of the sequence (p n ) n#N of best L 1  approximations of a polynomialtime 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
Proof Mining in L_1approximation (Extended Abstract)
"... 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 1approximation of continuous functions from the ine#ective proof of uniqueness due to Cheney. The ..."
Abstract
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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 1approximation of continuous functions from the ine#ective proof of uniqueness due to Cheney. The