Approximation of Chaotic Signals Using Quadratic and Cubic Fractal Interpolation Functions

Abstract

The fundamental aim of the paper is to propose the construction of quadratic and cubic fractal interpolation functions with nonzero vertical scaling factors, using the concept of Banach contraction. The constructed quadratic and cubic fractal interpolation functions have been implemented to approximate and integrate the one-dimensional discrete data points. As the first step towards the approximation and the derivation of the integration formula, the given discrete set of data has been assigned with quadratic and cubic iterated function systems, which are then used in defining the respective Hutchinson operators. The given data set is graphically approximated with the unique fixed point of these newly formulated operators via the Banach contraction principle. The paper then focuses on the derivation of the method of numerical integration for the data set using the constructed quadratic and cubic fractal interpolation functions. The numerical integration formula is defined based on the coefficients in the corresponding iterated function systems. Finally, the paper illustrates the performance of the linear, quadratic and cubic iterated function systems in approximating and integrating signals with chaotic behavior. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.