Convergence of Cubic Spline Super Fractal Interpolation Functions

TL;DR

This paper introduces Cubic Spline Super Fractal Interpolation Functions (SFIF) to model complex embedded structures and proves their convergence properties, enabling precise approximation of functions and derivatives with arbitrary accuracy.

Contribution

The paper presents a novel Cubic Spline SFIF framework and establishes its convergence rates for function and derivative approximation.

Findings

Convergence of SFIF to data functions at rate h^(2-e)

Approximation of derivatives with similar convergence rates

Any desired accuracy achievable in function approximation

Abstract

In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x_0,x_N], the interpolating Cubic Spline (SFIF) and their derivatives converge respectively to the data generating function and its derivatives at the rate of h^(2-j+e) (0<e<1), j=0,1,2 as the norm h of the partition of [x_0,x_N] approaches zero. The convergence results for Cubic Spline (SFIF) found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline (SFIF) and its corresponding derivatives.