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Fractal Based Rational Cubic Trigonometric Zipper Interpolation with Positivity Constraints

A. K. Sharma, K. R. Tyada

TL;DR

This work introduces Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs) to interpolate positive data while capturing irregular, fractal-like behavior. By embedding a zipper signature and using rational cubic trigonometric forcing terms within a Read-Bajraktarević-IFS framework, the authors achieve positivity preservation through explicit bounds on the vertical scaling factors and shape parameters, and establish convergence to the data-generating function via rigorous error analysis. Theoretical results include a concrete error bound that combines an epsilon-perturbation term, a cubic-term in the subinterval length, and a fractal-parameter term, ensuring stability as the signature deviation and step size shrink. Numerical experiments corroborate the positivity-preserving property and illustrate how parameter choices influence local curve shape and fractal irregularity, highlighting the practical value of RCTZFIFs for modeling non-smooth, positive data with controlled geometric features.

Abstract

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.

Fractal Based Rational Cubic Trigonometric Zipper Interpolation with Positivity Constraints

TL;DR

This work introduces Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs) to interpolate positive data while capturing irregular, fractal-like behavior. By embedding a zipper signature and using rational cubic trigonometric forcing terms within a Read-Bajraktarević-IFS framework, the authors achieve positivity preservation through explicit bounds on the vertical scaling factors and shape parameters, and establish convergence to the data-generating function via rigorous error analysis. Theoretical results include a concrete error bound that combines an epsilon-perturbation term, a cubic-term in the subinterval length, and a fractal-parameter term, ensuring stability as the signature deviation and step size shrink. Numerical experiments corroborate the positivity-preserving property and illustrate how parameter choices influence local curve shape and fractal irregularity, highlighting the practical value of RCTZFIFs for modeling non-smooth, positive data with controlled geometric features.

Abstract

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.

Paper Structure

This paper contains 7 sections, 5 theorems, 42 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries of Fractal Interpolation Functions
  3. Construction of Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs)
  4. Convergence Analysis
  5. Positivity preserving RCTZFIFs
  6. Numerical Experiments
  7. Conclusion

Key Result

Theorem 2.1

The IFS $\{W_i\}_{i=1}^{n-1}$ admits a unique attractor $G$, which coincides with the graph of a continuous function $h^* : I \rightarrow \mathbb{R}$ that interpolates the given data, i.e., $h^*(t_j) = f_j$ for each $j$. Moreover, let $\mathcal{C}(I) = \{h \in C(I) : h(t_1) = f_1,\ h(t_n) = f_n\}$ b Then $T$ has a unique fixed point $h^*$ in $\mathcal{C}(I)$, referred to as the fractal interpolati

Figures (1)

  • Figure 1: Visualisations of RCTZFIFs under different parameter configurations.

Theorems & Definitions (6)

  • Theorem 2.1: Existence of FIF, cf. BarHar86
  • Theorem 2.2: Existence of Rational Spline FIFs CK2015
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • proof