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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 ..."
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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].
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
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Approximation in Normed Linear Spaces
, 2000
"... 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 usu ..."
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Cited by 6 (0 self)
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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 ...
On the Computational Complexity of Best L1Approximation
, 2001
"... We give an upper bound on the complexity of the sequence (p n ) n#N of best L1approximations 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 L1approximation presen ..."
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Cited by 2 (1 self)
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We give an upper bound on the complexity of the sequence (p n ) n#N of best L1approximations 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 L1approximation presented in [KO01].
Mathematical Logic Quarterly c © WILEYVCH Verlag Berlin GmbH 2001 On the computational complexity of best L1approximation
"... Abstract. It is well known that for a given continuous function f: [0, 1] ! R and a number n there exists a unique polynomial pn 2 Pn (polynomials of degree n) which best L1approximates f. We establish the rst upper bound on the complexity of the sequence (pn)n2N, assuming f is polynomialtime com ..."
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Abstract. It is well known that for a given continuous function f: [0, 1] ! R and a number n there exists a unique polynomial pn 2 Pn (polynomials of degree n) which best L1approximates f. We establish the rst upper bound on the complexity of the sequence (pn)n2N, assuming f is polynomialtime computable. Our complexity analysis makes essential use of the modulus of uniqueness for L1approximation presented in [13].