Lidstone Fractal Interpolation and Error Analysis

TL;DR

This paper introduces Lidstone Fractal Interpolation Functions (Lidstone FIF), extending classical interpolation methods to better approximate real-world data, with proven existence, construction methods, and error estimates demonstrating their effectiveness.

Contribution

The paper develops the concept of Lidstone FIF, proves their existence, provides a computational construction method, and establishes error bounds for approximation accuracy.

Findings

Lidstone FIF exists for given data and is a $C^{2p}$ fractal function.

Error in function approximation is of order $N^{-2p}$.

Error in derivative approximation is of order $N^{-(2p-2k)}$.

Abstract

In the present paper, the notion of Lidstone Fractal Interpolation Function ($Lidstone \ FIF$) is introduced to interpolate and approximate data generating functions that arise from real life objects and outcomes of several scientific experiments. A Lidstone FIF extends the classical Lidstone Interpolation Function which is generally found not to be satisfactory in interpolation and approximation of such functions. For a data $\{(x_n,y_{n,2k}); n=0,1,\ldots,N \ \text{and} \ k=0,1,\ldots,p\}$ with $N,p\in\mathbb{N}$, the existence of Lidstone FIF is proved in the present work and a computational method for its construction is developed. The constructed Lidstone FIF is a $C^{2p}[x_0,x_N]$ fractal function $\ell_\alpha$ satisfying $\ell_\alpha^{(2k)}(x_n)=y_{n,2k}$, $n=0,1,\ldots,N$,\ $k=0,1,\ldots,p$. Our error estimates establish that the order of $L^\infty$-error in approximation of a data generating function in $C^{2p}[x_0,x_N]$ by Lidstone FIF is of the order $N^{-2p}$, while $L^\infty$-error in approximation of $2k$-order derivative of the data generating function by corresponding order derivative of Lidstone FIF is of the order $N^{-(2p-2k)}$. The results found in the present work are illustrated for computational constructions of a Lidstone FIF and its derivatives with an example of a data generating function.