bibkey: herrera2023functionapproximation
authors: D. Herrera
year: 2023
title: On the commutator modulus of continuity for operator monotone functions
doi: null
url: https://arxiv.org/abs/2311.17448v1
claim: Conjecture 2 asserts that E_f(c) ≤ C f(c) for every non-negative operator monotone f on [0,∞) and every c > 0 when C = 1.
strata_touched:
  - D5/S3/Fourier/PositiveFourierApproximationRefutation
license: citation-only
triage: anchor

Herrera’s function-approximation conjecture

The paper defines [ E_f(c)=\inf_{g’\in\mathcal M_+^\wedge} \left(\sup_{x\ge 0}(f-g)(x)-\inf_{x\ge 0}(f-g)(x)+cg’(0)\right), ] where (\mathcal M_+^\wedge) is the space of functions that are Fourier transforms of finite positive Borel measures on (\mathbb R). Section 5 uses the equivalent formulation in which (g) is (C^1(\mathbb R)) and (g’) has that Fourier-transform form.

The source states: “Let (f) be a non-negative function that is operator monotone on ([0, \infty)). Then (E_{f}(c) \leq Cf(c)) holds for all (c > 0) when (C = 1).” This is Conjecture 2, numbered 1.10 in print. The paper proves (E_f(c)\leq C f(c)) with (C<1.01975) for (f(x)=x/(x+1)) using Gaussian approximants.

The conjecture implies the paper’s Conjectures 1 and 3, which are commutator inequalities. A refutation of Conjecture 2 therefore removes that implication; the two commutator conjectures are not settled by this result and remain open here.

Verified locator

Source: https://arxiv.org/abs/2311.17448v1