Approximation by polynomials with restricted ranges of their derivatives

Roulier, John A.; Taylor, G. D.

doi:10.1016/0021-9045(72)90015-9

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Applications For C1 -Boundaries (L1-^-spaces)1

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1992

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Cited by 15 publications

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“…When aci = 0, uniqueness of a best constrained uniform approximation requires that the function / being approximated satisfy the constraint vx < f <ux (see [2,12]). This is not the case for L1-approximation.…”

Section: Applications For C1 -Boundaries (L1-^-spaces)mentioning

confidence: 99%

A Haar-type theory of best 𝐿₁-approximation with constraints

Kroó¹,

Schmidt²

1992

Trans. Amer. Math. Soc.

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Abstract.A general setting for constrained Z,1-approximation is presented. Let Un be a finite dimensional subspace of C [a, b] and L be a linear operator from Un to C(K) (r = 0, 1) where K is a finite union of disjoint, closed, bounded intervals. For v , u e C(K) with v < u, the approximating set is Univ, u) = {p e Un : v < Lp < u on K} and the norm is \\f\\w = Xf \f\wdx where w a positive continuous function on [a, b]. We obtain necessary and sufficient conditions for Un (v, u) to admit unique best || • ||w-approximations to all / € C[a, b] for all positive continuous w and all v , u € C(K) (r = 0, 1) satisfying a nonempty interior condition. These results are applied to several L1-approximation problems including polynomial and spline approximation with restricted derivatives, lacunary polynomial approximation with restricted derivatives, and others.

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Section: Applications For C1 -Boundaries (L1-^-spaces)mentioning

confidence: 99%