We propose a novel fractal based interpolation scheme termed Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs) designed to model and preserve the inherent geometric property, positivity, in given datasets. The method employs a combination of rational cubic trigonometric functions within a zipper fractal framework, offering enhanced flexibility through shape parameters and scaling factors. Rigorous error analysis is presented to establish the convergence of the proposed zipper fractal interpolants to the underlying classical fractal functions, and subsequently, to the data-generating function. We derive necessary constraints on the scaling factors and shape parameters to ensure positivity preservation. By carefully selecting the signature, shape parameters, and scaling factors within these bounds, we construct a class of RCTZFIFs that effectively preserve the positive nature of the data, as compared to a reference interpolant that may violate this property. Numerical experiments and visualisations demonstrate the efficacy and robustness of our approach in preserving positivity while offering fractal flexibility.

翻译：本文提出了一种新颖的基于分形的插值方案，称为有理三次三角Zipper分形插值函数（RCTZFIFs），旨在建模并保持给定数据集中固有的几何特性——正性。该方法在zipper分形框架内结合了有理三次三角函数，通过形状参数和缩放因子提供了更强的灵活性。我们给出了严格的误差分析，以证明所提出的zipper分形插值函数对基础经典分形函数的收敛性，进而对数据生成函数的收敛性。我们推导了缩放因子和形状参数的必要约束条件以确保正性保持。通过在这些界限内仔细选择特征、形状参数和缩放因子，我们构建了一类能有效保持数据正性的RCTZFIFs，相比之下，参考插值函数可能违反该性质。数值实验和可视化结果表明，我们的方法在保持分形灵活性的同时，能有效且稳健地保持正性。