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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].
Strong Uniqueness
, 2010
"... This is a survey paper on the subject of strong uniqueness in approximation theory. ..."
Abstract

Cited by 2 (2 self)
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This is a survey paper on the subject of strong uniqueness in approximation theory.
Exact Order of Hoffman's Error Bounds for Elliptic Quadratic Inequalities Derived From VectorValued Chebyshev Approximation
, 1999
"... In this paper, we introduce the exact order of Hoffman's error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vectorvalued functions (including complexvalued functions). The set of Chebyshev app ..."
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In this paper, we introduce the exact order of Hoffman's error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vectorvalued functions (including complexvalued functions). The set of Chebyshev approximations of a vectorvalued function defined on a finite set is shown to be Hausdorff strongly unique of order exactly 2 s for some nonnegative integer s. As a consequence, the exact order of Hoffman's error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 s for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated by the elliptic quadratic inequalities.
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 ..."
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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