关于二元Bernstein多项式插值的一个猜想

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2023-09-01 DOI:10.1016/j.jat.2023.105920

Michael S. Floater

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引用次数: 0

摘要

本文讨论了Schumacer的一个猜想，即三角网格上二元Bernstein多项式配置矩阵的主子矩阵具有正行列式。很容易证明这个猜想适用于2×2的子矩阵。在本文中，我们证明了它也适用于3×3子矩阵，与猜想的等价“单项式”一起工作。这一结果推广到一类3×。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On a conjecture concerning interpolation by bivariate Bernstein polynomials

In this paper we discuss a conjecture of Schumaker that the principal submatrices of collocation matrices of bivariate Bernstein polynomials over triangular grids have positive determinant. It is easy to show that the conjecture holds for the 2 × 2 submatrices. In this paper we show that it also holds for the 3 × 3 submatrices, working with the equivalent ‘monomial form’ of the conjecture. This result generalizes to a class of 3 × 3 matrices which will be described.

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.

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