Abstract:

Polynomial interpolation technique is a common approximation method in approximation theory, which is widely used in numerical analysis, signal processing, and so on. Traditional polynomial interpolation algorithms are mainly developed by combining numerical analysis with experimental results, lacking of unified theoretical description and regular solution. A uniform theoretical framework for polynomial interpolation algorithm based on osculating polynomial approximation theory was proposed here. Existing interpolation algorithms could be analyzed and new algorithms could be developed under this framework, which consists of the number of sample points, osculating order for sample points and derivative approximation rules. The presentation of existing mainstream interpolation algorithms was analyzed in proposed framework, and the general process for developing new algorithms was shown by using a four-point and two-order osculating polynomial interpolation. Theoretical analysis and numerical experiments show that almost all mainstream polynomial interpolation algorithms belong to osculating polynomial interpolation, and their effects are strongly related to the number of sampling points, order of osculating, and derivative approximation rules.

Key words: osculating polynomial, Derivative Approximation Rule (DAR), osculating order, polynomial interpolation, signal processing

摘要：

多项式插值技术是近似理论中一种常见的近似方法,被广泛用于数值分析、信号处理等领域。但传统的多项式插值技术大多是基于数值分析与实验结果相结合得到的,没有统一的理论描述和规律性的解决方案。为此,根据密切多项式近似理论为图像的多项式插值算法提出一个统一的理论框架。密切多项式近似的理论框架包括采样点数目、密切阶数和导数近似规则三个部分,它既可以用于分析现有的多项式插值算法,也可以用于开发新的多项式插值算法。分析了主流多项式插值技术在密切多项式近似理论框架下的表现形式,并以四点二阶密切多项式插值算法为例详细描述了利用密切多项式插值的理论框架开发新的多项式插值算法的一般流程。理论分析和数值实验表明大多数主流插值算法都属于密切多项式插值算法,它们的处理效果与采样点数目、密切阶数和导数近似规则有紧密的关系。

关键词: 密切多项式, 导数近似规则, 密切阶数, 多项式插值, 信号处理