Definite Integral of $\alpha$-Fractal Functions

TL;DR

This paper investigates the integration of $oldsymbol{ extalpha}$-fractal functions, providing estimates and relations between integrals of original and flipped fractal functions, despite the lack of explicit formulas.

Contribution

It establishes new results on the definite integral of $oldsymbol{ extalpha}$-fractal functions and introduces a relation between integrals of original and flipped fractal functions.

Findings

Derived estimates for the definite integral of $oldsymbol{ extalpha}$-fractal functions.

Constructed the flipped $oldsymbol{ extalpha}$-fractal function and related its integral to the original.

Provided theoretical relations between integrals of fractal functions and their flips.

Abstract

In this article the integration of the $\alpha$-fractal interpolation function $f^{\alpha}$ corresponding to any continuous function $f$ on a compact interval $I$ of $\mathbb{R}$ is estimated although there is no explicit form of $\alpha$-fractal interpolation function till now. Some results related to the definite integral of $f^{\alpha}$ are established. Also the flipped $\alpha$-fractal function $f_{F}^{\alpha_{F}}$ corresponding to the continuous function $f$ is constructed and a result is proved that relates the definite integrals of the fractal functions $f_{F}^{\alpha_{F}}$ and $f^{\alpha}$.