广义多元分形插值函数与α-分形函数
Generalized multivariate Fractal Interpolation Function and $α$-Fractal Function
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中文总结 AI 辅助
本文提出基于Matkowski和Rakotch压缩映射的新方法，推广多元分形插值函数的已有构造技术，建立非线性迭代函数系统吸引子为插值理论数据点的多元函数图像，并探讨Rakotch压缩下的不变Borel概率测度存在性。
中文摘要 AI 辅助
本文提出一种构造多元分形插值函数及与多元函数关联的α-分形函数的新方法。与现有依赖Banach压缩原理的方法不同，该构造基于Matkowski和Rakotch压缩映射。尽管文献中已探索了多种构造多元分形插值函数的方法，但本文提出的方法具有独特性，它推广了所有已知技术并为此类构造提供了更宽泛的框架。我们提出一种利用广义压缩映射构造非线性迭代函数系统的技术，证明这类系统的吸引子是插值理论数据点的连续多元函数的图像。此外，针对Rakotch压缩映射，本文探讨了支撑于关联多元分形插值函数图像上的不变Borel概率测度的存在性。
英文摘要
In this paper, we introduce a new approach for constructing multivariate fractal interpolation functions and $α$-fractal functions associated with multivariate functions. Unlike the existing methods that rely on the Banach contraction principle, here the construction is based on Matkowski and Rakotch contractions. While numerous methods for constructing multivariate fractal interpolation functions have been explored in the literature, the approach given in this paper is distinct in the sense that it generalizes all previously known techniques and provides a broader framework for such constructions. We propose a technique to develop nonlinear iterated function systems using the generalized contractions and establish that the attractors of such systems are the graphs of continuous multivariate functions interpolating theoretical data points. Furthermore, for the Rakotch contractions, the existence of an invariant Borel probability measure supported on the graph of the associated multivariate fractal interpolation function is explored.
