Rate of Convergence of the Linear Discrete Polya Algorithm

Quesada, José M.; Martínez-Moreno, Juan; Navas, J.; Fernández-Ochoa, J.

doi:10.1006/jath.2000.3552

Cited by 7 publications

(7 citation statements)

References 6 publications

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“…Moreover, in [8,10] it is deduced that there are constants M 1 , M 2 > 0 and 0 a 1, depending on K, such that…”

Section: This Condition Can Be Writtenmentioning

confidence: 98%

“…where h * ∞ is a particular best uniform approximation of 0 from K, called the strict uniform approximation [6,10] and whose definition is also valid in any closed, convex set K. The strict uniform approximation is determined by the next property. Let H denote the set of the best uniform approximations of 0 from K. For every h ∞ ∈ H we consider the vector (h ∞ ) whose coordinates are given by |h ∞ (j )|, 1 j n, arranged in decreasing order.…”

Section: This Condition Can Be Writtenmentioning

confidence: 99%

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Rate of convergence of the Pólya algorithm from polyhedral sets

Huotari

Marano

Navas

et al. 2005

Journal of Approximation Theory

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In this paper we consider a problem of best approximation in p , 1 < p ∞. Let h p denote the best p-approximation of h ∈ R n from a closed, convex set K of R n , 1 < p < ∞, h / ∈ K, and let h * ∞ be the strict uniform approximation of h from K. We prove that if K satisfies locally a geometrical property, fulfilled by any polyhedral set of R n , then lim sup p→∞ p h p − h * ∞ < ∞.

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“…Moreover, in [8,10] it is deduced that there are constants M 1 , M 2 > 0 and 0 a 1, depending on K, such that…”

Section: This Condition Can Be Writtenmentioning

confidence: 98%

Section: This Condition Can Be Writtenmentioning

confidence: 99%

Rate of convergence of the Pólya algorithm from polyhedral sets

Huotari

Marano

Navas

et al. 2005

Journal of Approximation Theory

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“…In this paper it is showed that p h p − h * ∞ is bounded. Subsequently, in [7] the authors established necessary and sufficient conditions on K to get that p h p − h * ∞ → 0 as p → ∞ and in [8] it is proved that there are constants L 1 , L 2 > 0 and 0 a 1, depending on K, such that…”

Section: Introductionmentioning

confidence: 99%

The Polya algorithm in sequence spaces

Quesada

Fernández-Ochoa

Martínez-Moreno

et al. 2005

Journal of Approximation Theory

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In this paper we consider the problem of best approximation in p (N), 1 < p ∞. If h p , 1 < p < ∞ denotes the best p -approximation of the element h ∈ 1 (N) from a finite-dimensional affine subspacewhere h * ∞ is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to give a complete description of the rate of convergence of h p − h * ∞ as p → ∞ by proving that there are constants L 1 , L 2 > 0 and 0 a 1 such thatfor p large enough. (J.M. Quesada), jmmoreno@ujaen.es (J. Martínez-Moreno), jbusta@fcm.uap.mx (J. Bustamante).

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“…Moreover, there exists a unique best ' p -approximation if K is a closed convex set and 15p51: In general, the unicity of the best ' 1 -approximation is not guaranteed. Throughout this paper, K denotes a proper affine subspace of R n and we will assume, without loss of generality, that h ¼ 0 and 0 = 2 K: In this context, we gave in [5] a complete description of the rate of convergence of the P ! o olya algorithm as p !…”

Section: Introductionmentioning

confidence: 99%