Polynomial Interpolation and a Multivariate Analog of Fundamental Theorem of Algebra
H. Hakopian, M. Tonoyan
本文从单变量多项式插值出发，将代数基本定理推广至特定多变量代数方程组系统。
从单变量多项式插值出发，我们得到了某些多变量代数方程组系统下代数基本定理的自然推广。
Starting with univariate polynomial interpolation we arrive to a natural generalization of fundamental theorem of algebra for certain systems of multivariate algebraic equations.
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An Algorithm on Multivariate Interpolation for Cartesian Grid Complex Boundary Problems
本文提出一种简单算法用于Cartesian网格上的多变量插值样条选择，通过嵌入边界方法解决椭圆界面问题验证了其有效性。
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In this paper, we study how to quickly compute the <-minimal monomial interpolating basis for a multivariate polynomial interpolation problem. We address the notion of "reverse" reduced basis of linearly independent polynomials and design an algorithm for it. Based on the notion, for any monomial ordering we present a new method to read off the <-minimal monomial interpolating basis from monomials appearing in the polynomials representing the interpolation conditions.
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We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the function has Lipschitz continuous gradient and is interpolated on an affinely independent sample set. Errors for quadratic functions and errors of bivariate linear extrapolation are analyzed in depth.
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本文刻画了重数之和至多为$2n+2$时Hermite多元插值问题的可解性集合，将作者2000年关于$2n+1$情形的定理进一步扩展，而后者可视为Severi (1921)经典定理的自然推广。
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本文探讨了最大体积原理在选择多项式基函数以确保拉格朗日插值问题有合理插值误差的应用，证明了勒贝格常数可通过体积和最小奇异值的倒数有效界束。
The natural kinship between classical theories of interpolation and approximation is well explored. In contrast to this, the interrelation between interpolation and approximation is subtle and this duality is relatively obscure in the context of fractal interpolation. The notion of $α$-fractal function provides a proper foundation for the approximation theoretic facet of univariate fractal interpolation functions (FIFs). However, no comparable approximation theoretic aspects of FIFs has been developed for functions of several variables. The current article intends to open the door for intrigui…
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Multivariate interpolation
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本文提出一个统一框架，将Padua点、Morrow-Patterson-Xu点和利萨如节点点的非张量多项式插值方案整合为一种理论，通过利萨如曲线和切比雪夫流形定义一般利萨如-切比雪夫节点，并推导离散正交结构，用于证明插值的唯一性和构造相应的求积法则。
We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions $\widetildeφ_j$ and trigonometric polynomials $φ_j$. The class of such operators includes classical interpolation polynomials ($\widetildeφ_j$ is the Dirac delta function), Kantorovich-type operators ($\widetildeφ_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on $\widetildeφ_j$ and $φ_j$, we obtain upper and lower bound estimates for the $L_p$-error of a…
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In this paper, we propose two methods for multivariate Hermite interpolation of manifold-valued functions. On the one hand, we approach the problem via computing suitable weighted Riemannian barycenters. To satisfy the conditions for Hermite interpolation, the sampled derivative information is converted into a condition on the derivatives of the associated weight functions. It turns out that this requires the solution of linear systems of equations, but no vector transport is necessary. This approach treats all given sample data points equally and is intrinsic in the sense that it does not dep…
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