Impact of End Point Conditions on the Representation and Integration of Fractal Interpolation Functions and Well Definiteness of the Related Operator

TL;DR

This paper investigates how endpoint conditions influence the graphical representation and numerical integration of fractal interpolation functions, addressing continuity issues and proposing recursive methods for approximation.

Contribution

It explores the impact of endpoint conditions on fractal interpolation functions and introduces a straightforward recursive method for their approximation.

Findings

Endpoint conditions significantly affect the graphical representation.

Continuity of fractal interpolation functions can be ensured through specific techniques.

Recursive relations are effective for approximating fractal interpolation functions.

Abstract

Fractal interpolation technique is an alternative to the classical interpolation methods especially when a chaotic signal is involved. The logic behind the formulation of an iterated function system for the construction of fractal interpolation functions is to divide the entire interpolating domain into subdomains and define functions on each subdomain piecewisely. The objective of this paper is to explore the significance of the end point conditions on the graphical representation of the resultant functions and their numerical integration. The central problem in the formulation of the IFS, the continuity of the fractal interpolation functions, is addressed with an explanation on the techniques implemented to resolve the problem. Instead of an analytical expression, the fractal interpolation functions are always represented in terms of recursive relations. This paper further presents the derivation of these recursive relations and proposes a straightforward method to find the approximating function, involved in these relations.