A Generalization of the GGR Conjecture

TL;DR

This paper presents a new proof of the GGR Theorem, a fundamental result linking Peano differentiability to generalized Riemann derivatives, and introduces a broader framework with multiple new smoothness conditions.

Contribution

The authors provide a novel proof of the GGR Theorem and generalize it to generate many new sets of Riemann smoothness conditions for differentiability.

Findings

New proof of the GGR Theorem established.

Generalization yields numerous new smoothness conditions.

Enhanced understanding of differentiability criteria through generalized Riemann derivatives.

Abstract

For each positive integer $n$, function $f$, and point $c$, the GGR Theorem states that $f$ is $n$ times Peano differentiable at $c$ if and only if $f$ is $n-1$ times Peano differentiable at $c$ and the following $n$-th generalized Riemann~derivatives of $f$ at $c$ exist: \[ \lim_{h\rightarrow 0}\frac 1{h^{n}}\sum_{i=0}^n(-1)^i\binom{n}{i}f(c+(n-i-k)h), \] for $k=0,\ldots,n-1$. The theorem has been recently proved in [AC2] and has been a conjecture by Ghinchev, Guerragio, and Rocca since 1998. We provide a new proof of this theorem, based on a generalization of it that produces numerous new sets of $n$-th Riemann smoothness conditions that can play the role of the above set in the GGR Theorem.