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Comonotone approximation and interpolation by entire functions II

Maxim R. Burke

TL;DR

This paper extends Hoischen-type approximation to simultaneous comonotone interpolation of a $C^n$ function and its derivatives by an entire function. It develops a formal $(Q_n)$ framework, proves half-line polynomial approximations that fix high-order endpoint data, and then assembles a global comonotone interpolant under the assumption that $(Q_n)$ holds. The authors establish the $n\le 3$ case of $(Q_n)$ (building on prior work), and derive a constructive theorem guaranteeing an entire $g$ that matches $f$ and its derivatives on a discrete set $E$ while maintaining comonotonicity of all derivatives up to order $n$. Collectively, these results advance comonotone approximation theory by linking Hoischen-type approximants to structured, monotonicity-preserving interpolation on the real line.

Abstract

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be approximated by an entire function $g$ so that for $k=0,\dots,n$, and $x\in\mathbb{R}$, $|D^{k}g(x)-D^{k}f(x)|<\varepsilon(x)$, and if $x\in E$ then $D^{k}g(x)=D^{k}f(x)$. The approximating function $g$ is entire and hence piecewise monotone. Building on earlier work, for $n\leq 3$, we determine conditions under which when $f$ is piecewise monotone we can choose $g$ to be comonotone with $f$ (increasing and decreasing on the same intervals), and under which the derivatives of $g$ can be taken to be comonotone with the corresponding derivatives of $f$ if the latter are piecewise monotone. The proof for $n\leq 3$ establishes the theorem for all $n$, assuming a conjecture (shown in previous work with Haris and Madhavendra to hold for $n\leq 3$) regarding the set of $2(n+1)$-tuples $(f(0),Df(0),\dots,D^nf(0),f(1),Df(1),\dots,D^nf(1))$ of the values at the endpoints of the derivatives of a $C^n$ function $f$ on $[0,1]$ for which $D^nf$ is increasing and not constant.

Comonotone approximation and interpolation by entire functions II

TL;DR

This paper extends Hoischen-type approximation to simultaneous comonotone interpolation of a function and its derivatives by an entire function. It develops a formal framework, proves half-line polynomial approximations that fix high-order endpoint data, and then assembles a global comonotone interpolant under the assumption that holds. The authors establish the case of (building on prior work), and derive a constructive theorem guaranteeing an entire that matches and its derivatives on a discrete set while maintaining comonotonicity of all derivatives up to order . Collectively, these results advance comonotone approximation theory by linking Hoischen-type approximants to structured, monotonicity-preserving interpolation on the real line.

Abstract

A theorem of Hoischen states that given a positive continuous function , an integer , and a closed discrete set , any function can be approximated by an entire function so that for , and , , and if then . The approximating function is entire and hence piecewise monotone. Building on earlier work, for , we determine conditions under which when is piecewise monotone we can choose to be comonotone with (increasing and decreasing on the same intervals), and under which the derivatives of can be taken to be comonotone with the corresponding derivatives of if the latter are piecewise monotone. The proof for establishes the theorem for all , assuming a conjecture (shown in previous work with Haris and Madhavendra to hold for ) regarding the set of -tuples of the values at the endpoints of the derivatives of a function on for which is increasing and not constant.
Paper Structure (4 sections, 24 theorems, 34 equations)

This paper contains 4 sections, 24 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. The statement $(Q_n)$
  3. Approximating polynomials on a half-line
  4. Simultaneous comonotone approximation of a $C^n$ function and its derivatives

Key Result

Theorem 1.1

Ho1975 Let $n$ be a nonnegative integer. Let $E\subseteq \mathbb{R}$ be a closed discrete set. Suppose $f\colon\mathbb{R}\to\mathbb{R}$ is a $C^n$ function and ${\varepsilon}\colon\mathbb{R}\to\mathbb{R}$ is a positive continuous function. Then there exists an entire function $g$ such that $g(\mathb

Theorems & Definitions (57)

  • Theorem 1.1
  • Proposition 1.2: Bu2019, Proposition 2.1
  • Definition 1.3: Bu2019, Definition 2.4
  • Proposition 1.4: Bu2019, Propositions 2.5, 2.6, 2.7
  • Remark 1.5
  • Proposition 1.6: Bu2019, Proposition 2.8
  • Proposition 1.7: Bu2019, Proposition 2.9
  • Proposition 1.8
  • proof
  • Proposition 1.9
  • ...and 47 more