Counterexamples to the Gaussian vs. MZ derivatives Conjecture

TL;DR

This paper provides counterexamples to a conjecture that Gaussian derivatives are the only MZ derivatives, showing that other types of generalized Riemann derivatives also qualify, thus broadening the understanding of derivative equivalences.

Contribution

The paper disproves the conjecture that Gaussian derivatives are the only MZ derivatives by constructing explicit counterexamples, expanding the classification of generalized Riemann derivatives.

Findings

Counterexamples invalidate the conjecture.

Gaussian derivatives are not the only MZ derivatives.

The classification of derivatives is expanded.

Abstract

J. Marcinkiewicz and A. Zygmund proved in 1936 that the special $n$-th generalized Riemann derivative ${_2}D_nf(x)$ with nodes $0,1,2,2^2,\ldots, 2^{n-1}$, is equivalent to the $n$-th Peano derivative $f_{(n)}(x)$, for all $n-1$ times Peano differentiable functions $f$ at~$x$. Call every $n$-th generalized Riemann derivative with this property an MZ derivative. The recent paper Ash, Catoiu, and Fejzi\'c [Israel J. Math. {255} (2023):177--199] introduced the $n$-th Gaussian derivatives as the $n$-th generalized Riemann derivatives with nodes either $0,1,q,q^2,\ldots ,q^{n-1}$ or $1,q,q^2,\ldots ,q^{n}$, where~$q\neq0,\pm 1$, proved that the Gaussian derivatives are MZ derivatives, and conjectured that these are \emph{all} MZ derivatives. In this article, we invalidate this conjecture by means of two counterexamples. The order in which these are presented allows an update of the conjecture after each counterexample. The proof of the first counterexample is simple, by scales of generalized Riemann derivatives. The proof of the second involves the classification of generalized Riemann derivatives of Ash, Catoiu, and Chin [Proc. Amer. Math. Soc {146} (2018):3847--3862]. Symmetric versions of all the results are also~included.