Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Behaviour Preserving Extensions of Univariate and Bivariate Functions

David Levin

TL;DR

The paper addresses behavior-preserving extensions of univariate and bivariate functions from a domain $D_0$ to a larger domain $D$ by leveraging a unified linear-model framework with constant or varying coefficients. It develops univariate constant-coefficient extensions, a general varying-coefficient approach with a smoothing functional, and a 2-D extension method that enforces smoothness via a biharmonic-like operator, all demonstrated on noisy data. It then introduces an efficient spline-based, parameter-free scheme using model-splines and a basis that inherently satisfies the chosen model, plus techniques to interpolate between models to blend behaviors. Overall, the work provides both principled extension algorithms and practical, scalable tools (including model-spline bases) for preserving boundary behavior and global trends in noisy data, with implications for signal processing and numerical extension tasks.

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.

Behaviour Preserving Extensions of Univariate and Bivariate Functions

TL;DR

The paper addresses behavior-preserving extensions of univariate and bivariate functions from a domain to a larger domain by leveraging a unified linear-model framework with constant or varying coefficients. It develops univariate constant-coefficient extensions, a general varying-coefficient approach with a smoothing functional, and a 2-D extension method that enforces smoothness via a biharmonic-like operator, all demonstrated on noisy data. It then introduces an efficient spline-based, parameter-free scheme using model-splines and a basis that inherently satisfies the chosen model, plus techniques to interpolate between models to blend behaviors. Overall, the work provides both principled extension algorithms and practical, scalable tools (including model-spline bases) for preserving boundary behavior and global trends in noisy data, with implications for signal processing and numerical extension tasks.

Abstract

Given function values on a domain , possibly with noise, we examine the possibility of extending the function to a larger domain , . In addition to smoothness at the boundary of , the extension on should also inherit behavioral trends of the function on , 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.

Paper Structure

This paper contains 11 sections, 2 theorems, 37 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Univariate case - from a linear model to extension
  3. Extracting linear prediction models
  4. Approximation and extension algorithms
  5. Univariate models with constant coefficients
  6. Univariate models with varying coefficients
  7. The scope of linear models with varying coefficients
  8. The bivariate case - from a linear model to smooth extension
  9. Efficient approximation-extension using spline basis functions model
  10. Model-Spline basis functions for approximation-extension
  11. Interpolation between models

Key Result

Proposition 5.1

Let $P=\{p_{k,\ell}\}_{(k,\ell)\in K}$ represent a linear model with constant coefficients, where $K$ is a finite subset of $\mathbb{Z}^2$, and let $\phi$ be a function of compact support in $\mathbb{R}^2$. Consider where $\{c_{i,j}\}_{(i,j)\in \mathbb{Z}^2}$ is a bi-infinite sequence satisfying the model, i.e., Then, the function $g$ satisfies the model, i.e.,

Figures (19)

  • Figure 1: Approximation to non-noisy data on $[0,7]$ and extension on $[0,14]$
  • Figure 2: Approximation to noisy data on $[0,7]$ and extension on $[0,14]$
  • Figure 3: Approximation to the non-noisy data on $[0,7]$.
  • Figure 4: Approximation-extension on $[0,14]$
  • Figure 5: Rational coefficients model: Approximation to the non-noisy data on $[0,7]$.
  • ...and 14 more figures

Theorems & Definitions (4)

  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof