一类新的单调/凸有理分形函数
A New Class of Monotone/Convex Rational Fractal Function
摘要：
本文给出了一种隐式定义的有理三次样条分形插值函数（RCSFIF）的描述和分析，该函数在每个子区间内具有两个形状参数。为了更准确地说，我们考虑迭代函数系统（IFS）与$q_n=\frac{P_n}{Q_n}$，$n \in \mathbb{N}_{N-1}$，其中$P_n(x)$是三次多项式，通过相应的FIF和$Q_n(x)$的插值条件是预先指定的二次多项式，每个包含两个自由的形状/合理性参数。我们确定了所提出的RCSFIF $g$相对于统一范数收敛于原始函数$Φ\in \mathcal{C}^3(I)$。我们还提供了充分条件的自动选择的合理IFS参数，以保持单调性和凸性的一组指定的数据点。我们考虑一些例子来说明发展的分形插值方案及其形状保持方面。
英文摘要：
This paper presents a description and analysis of a rational cubic spline FIF (RCSFIF) that has two shape parameters in each subinterval when it is defined implicitly. To be precise, we consider the iterated function system (IFS) with $q_n=\frac{P_n}{Q_n}$, $n \in \mathbb{N}_{N-1}$, where $P_n(x)$ are cubic polynomials to be determined through interpolatory conditions of the corresponding FIF and $Q_n(x)$ are preassigned quadratic polynomials each containing two free shape/rationality parameters. We establish the convergence of the proposed RCSFIF $g$ to the original function $Φ\in \mathcal{C}^3(I)$ with respect to the uniform norm. We also provide the sufficient conditions for an automatic selection of the rational IFS parameters to preserve monotonicity and convexity of a prescribed set of data points. We consider some examples to illustrate the developed fractal interpolation scheme and its shape preserving aspects.
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