Behaviour Preserving Extensions of Univariate and Bivariate Functions

TL;DR

This paper investigates methods for extending functions from a smaller domain to a larger one while preserving their behavioral characteristics like growth, decay, or oscillations, using linear models with adaptable coefficients.

Contribution

It introduces a novel approach for behavior-preserving function extension based on linear models with variable coefficients, ensuring smoothness and trend inheritance.

Findings

Effective extension methods that preserve growth and oscillation patterns.

Framework applicable to noisy and smooth function data.

Potential for broad application in data interpolation and modeling.

Abstract

Given function values on a domain $D_0$, possibly with noise, we examine the possibility of extending the function to a larger domain $D$, $D_0\subset D$. In addition to smoothness at the boundary of $D_0$, the extension on $D\setminus D_0$ should also inherit behavioral trends of the function on $D_0$, such as growth and decay or even oscillations. The approach chosen here is based upon the framework of linear models, univariate or bivariate, with constant or varying coefficients.