Lipschitz conditions on uniform approximation operators

Cline, Alan

doi:10.1016/0021-9045(73)90025-7

Cited by 68 publications

(21 citation statements)

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“…For other information on Lipschitz conditions for the Chebyshev projection, see [1,7,8,9]. \\P(a) -P(a')\\ ^ 2AMe)( T a(e)N+i~l -l)\\a -a'll < e.…”

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confidence: 99%

Lipschitz Constants and Moduli of Continuity for the Chebyshev Projection

Bridges¹

1982

Proceedings of the American Mathematical Society

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Abstract. It is shown how to construct Lipschitz constants and moduli of continuity for the Chebyshev projection of C[0,1] onto the finite-dimensional subspace spanned by a Chebyshev system.It is well known that if X is a finite-dimensional linear subspace of a real normed space E, and if each a G E has a unique best approximant P(a) in X, then the projection P of E onto X is continuous (cf. [6, pp. 25-26]). We shall discuss this result in the context of Chebyshev approximation over [0,1].Although our discussion will take place entirely within the framework of Bishop's constructive mathematics [2,3], the results and estimates obtained below are also new to classical (that is, traditional) mathematics. Indeed, it is difficult to see how these estimates could have been obtained by someone working outside the constructive framework.Let

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“…For other information on Lipschitz conditions for the Chebyshev projection, see [1,7,8,9]. \\P(a) -P(a')\\ ^ 2AMe)( T a(e)N+i~l -l)\\a -a'll < e.…”

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confidence: 99%

Lipschitz Constants and Moduli of Continuity for the Chebyshev Projection

Bridges¹

1982

Proceedings of the American Mathematical Society

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“…Если f ∈ M , то, оче-видно, γ(f ) = 1. Точный результат содержится в [2], [3], где для f ∈ C(Q) \ M и произвольного конечномерного чебышевского подпространства M доказана формула γ(f ) = min ϕ∈M ∥ϕ∥=1…”

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“…Неравенство γ(C(Q)) > 0 выполняется только в двух случаях: если M одно-мерно [18] или Q -конечное множество [2], [3]. В иных случаях γ(B(0, 1)) = 0 [19], здесь же можно найти обобщение этого результата на произвольные компакты Q).…”

Section: а в мариновunclassified

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О Равномерных Константах Сильной Единственности В Чебышевских Приближениях И Основополагающих Результатах Н. Г. Чеботарева

Marinov¹

2011

Izv. RAN. Ser. Mat.

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“…Freud's proof may be conceived as a (partial) proof-analysis in our sense. Although Freud himself does not exhibit the numerical content of his proof one can obtain the estimate of [11] (mentioned in 2.) for the Lipschitz constant quite easily from his proof.…”

Section: By 32 and The Definition Of χ It Follows Thatmentioning

confidence: 99%

New effective moduli of uniqueness and uniform a priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory

Kohlenbach

1993

Numerical Functional Analysis and Optimization

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Let U and V be complete separable metric spaces, Vu compact in V and G : U × V → IR a continuous function. For a large class of (usually non-constructive) proofs of uniqueness theoremsone can extract an effective modulus of uniqueness Φ by logical analysis, i.e.Since Φ does not depend on v1, v2 it is an a-priori estimate, which generalizes the notion of strong unicity in Chebycheff approximation theory. This applies to uniqueness proofs in Chebycheff approximation, · 1-approximation of f ∈ C[0, 1] and best uniform approximation by polynomials having bounded coefficients. Here we continue our proof-theoretic analysis started in [21]. A simplification of a proof by Young/Rice and a variant due to Borel are analysed yielding explicit moduli Φ and uniform a-priori (lower) estimates for strong unicity which are significantly better than those obtained from de La Vallée Poussin's uniqueness proof in [21]. It is explained how the numerical content of the three proofs depends on the logical form whereby certain analytical lemmas (e.g. the alternation theorem) play a key-role. The numerical results obtained from Young/Rice's proof are best and substantially improve known estimates obtained by D. Bridges by an n (=dimension of the Haar space) in the exponent.

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Lipschitz conditions on uniform approximation operators

Cited by 68 publications

References 2 publications

Lipschitz Constants and Moduli of Continuity for the Chebyshev Projection

Lipschitz Constants and Moduli of Continuity for the Chebyshev Projection

О Равномерных Константах Сильной Единственности В Чебышевских Приближениях И Основополагающих Результатах Н. Г. Чеботарева

New effective moduli of uniqueness and uniform a priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory

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