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Approximation by polynomials with only real critical points

David L. Bishop

TL;DR

The paper strengthens the Weierstrass approximation theorem by showing that any real-valued continuous function on a compact interval can be uniformly approximated by polynomials whose critical points all lie inside the interval. The core method combines a carefully designed perturbation of Chebyshev polynomials with a Brouwer fixed point construction to force the derivatives to realize prescribed nodal averages, yielding density of CP-constrained polynomials in $C_\mathbb{R}(I)$ (with Lipschitz control when $f$ is Lipschitz). It also establishes a weak-* approximation result for bounded functions by polynomials with only real zeros, and analyzes the asymptotic behavior of the derivatives of these approximants, proving they diverge almost everywhere in many cases. Finally, it identifies limitations by showing that the corresponding CP(X) property can fail on certain disconnected subsets of $\mathbb{R}$, highlighting the nuanced dependence on the geometry of the underlying set.

Abstract

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval $I \subset \mathbb{R}$ can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical points are contained in $I$. The proof uses a perturbed version of the Chebyshev polynomials and an application of the Brouwer fixed point theorem.

Approximation by polynomials with only real critical points

TL;DR

The paper strengthens the Weierstrass approximation theorem by showing that any real-valued continuous function on a compact interval can be uniformly approximated by polynomials whose critical points all lie inside the interval. The core method combines a carefully designed perturbation of Chebyshev polynomials with a Brouwer fixed point construction to force the derivatives to realize prescribed nodal averages, yielding density of CP-constrained polynomials in (with Lipschitz control when is Lipschitz). It also establishes a weak-* approximation result for bounded functions by polynomials with only real zeros, and analyzes the asymptotic behavior of the derivatives of these approximants, proving they diverge almost everywhere in many cases. Finally, it identifies limitations by showing that the corresponding CP(X) property can fail on certain disconnected subsets of , highlighting the nuanced dependence on the geometry of the underlying set.

Abstract

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical points are contained in . The proof uses a perturbed version of the Chebyshev polynomials and an application of the Brouwer fixed point theorem.
Paper Structure (13 sections, 36 theorems, 138 equations, 14 figures)

This paper contains 13 sections, 36 theorems, 138 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Estimating the length of the nodal intervals
  3. The area of a Chebyshev node
  4. Perturbing the roots
  5. The distortion of 2-point perturbations
  6. The distortion of 3-point perturbations
  7. Bounding the extreme values
  8. Estimating the effect of interior perturbations
  9. Estimating the effect of exterior perturbations
  10. Applying Brouwer's fixed point theorem
  11. Weak-$\ast$ convergence: Proof of Theorem \ref{['weak limit']}
  12. Divergence almost everywhere
  13. Theorem \ref{['weier++']} fails for some subsets of $\mathbb R$

Key Result

Theorem 1.1

Suppose $f:I \to \mathbb R$ is a continuous function on a compact interval $I \subset \mathbb R$. Then for any $\epsilon>0$, there is a real polynomial $p$ so that $\lVert f-p\rVert_I < \epsilon$ and ${\rm{CP}}(p) := \{z \in \mathbb C : p'(z) =0\} \subset I$, i.e., every real or complex critical po

Figures (14)

  • Figure 1: A 2-point perturbation of $T_{33}$. The left picture shows all of $[-1,1]$ and the right shows an enlargement of the interval where the perturbation occurs. The Chebyshev polynomial is solid and the perturbation is dashed. The white dots are the two new root locations.
  • Figure 2: A 3-point perturbation. The top figure shows $T_{33}$ (solid) and the perturbation $\widetilde{T}_n$ (dashed) on $[-1,1]$. The bottom figure is an enlargement around the perturbed roots.
  • Figure 3: On top we have perturbed $T_{33}$ (dashed) to obtain $p'$ (solid): four pairs in $[0,1]$ chosen to make the function more positive, and four pairs in $[-1,0]$ chosen to make it more negative. The bottom picture shows $p = \int p'$ (solid), which approximates $f(x) = |x|$ (dashed). See Figure \ref{['Approx_ABS_deg201']} for a higher degree approximation.
  • Figure 4: On the left is a degree 201 polynomial approximating $|x|$. The right picture is a log-log plot of the sup-norm difference between $f(x)=|x|$ and our approximation for degrees between 200 and 1000. The best linear fit is $\approx (-.9912) t+1.8972$. The optimal polynomial approximations (no restrictions) behave like $\approx -t-1.2724$.
  • Figure 5: On the left is the perturbed Chebyshev polynomial of degree 100, and on the right is its integral. From the picture it seems clear that any Lipschitz function can be approximated; the goal of the paper is to prove this is correct.
  • ...and 9 more figures

Theorems & Definitions (60)

  • Theorem 1.1: Critically Constrained Weierstrass Theorem
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 50 more