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A conjecture on multivariate polynomial interpolation.

Jesús Miguel Carnicer, Mariano Gasca (2001)

RACSAM

La generalización de las fórmulas de interpolación de Lagrange y Newton a varias variables es uno de los temas habituales de estudio en interpolación polinómica. Dos clases de configuraciones geométricas particularmente interesantes en el plano fueron obtenidas por Chung y Yao en 1978 para la fórmula de Lagrange y por Gasca y Maeztu en 1982 para la de Newton. Estos últimos autores conjeturaron que toda configuración de la primera clase es de la segunda, y probaron que el recíproco no es cierto....

A counterexample in comonotone approximation in L p space

Xiang Wu, Song Zhou (1993)

Colloquium Mathematicae

Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function f ∈ C [ - 1 , 1 ] k , with f ( k ) ( x ) ≥ 0 for x ∈ [0,1] and f ( k ) ( x ) ≤ 0 for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where e n ( k ) ( f ) p is the best approximation of degree n to f in L p by polynomials which are comonotone with f, that is, polynomials P so that P ( k ) ( x ) f ( k ) ( x ) ≥ 0 for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution...

A Dichotomy Principle for Universal Series

V. Farmaki, V. Nestoridis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ( α j ) j = 1 ∞ of scalars, there exists a subsequence ( α k j ) j = 1 ∞ such that either every subsequence of ( α k j ) j = 1 ∞ defines a universal series, or no subsequence of ( α k j ) j = 1 ∞ defines a universal series. In particular examples we decide which of the two cases holds.

A fast iteration for uniform approximation

Ferenc Kálovics (1988)

Aplikace matematiky

The paper gives such an iterative method for special Chebyshev approxiamtions that its order of convergence is ≥ 2 . Somewhat comparable results are found in [1] and [2], based on another idea.

A function related to a Lagrange-Bürmann series

Paul Bracken (2002)

Czechoslovak Mathematical Journal

An infinite series which arises in certain applications of the Lagrange-Bürmann formula to exponential functions is investigated. Several very exact estimates for the Laplace transform and higher moments of this function are developed.

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