Analyse fonctionnelle

A new proof of the GGR conjecture

Ash, J. Marshall¹ ; Catoiu, Stefan¹  ; Fejzić, Hajrudin²

¹ Department of Mathematics, DePaul University, Chicago, IL 60614
² Department of Mathematics, California State University, San Bernardino, CA 92407

Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 349-353

Résumé

For each positive integer $n$, function $f$, and point $x$, the 1998 conjecture by Ginchev, Guerragio, and Rocca states that the existence of the $n$th Peano derivative $f_{\left(n\right)}\left(x\right)$ is equivalent to the existence of all $n(n+1)/2$ generalized Riemann derivatives,

$$D_{k,-j}f\left(x\right)=\underset{h\to 0}{lim}\frac{1}{h^n}\sum _{i=0}^k{(-1)}^i\left(\genfrac{}{}{0pt}{}{k}{i}\right)f(x+(k-i-j)h),$$

for $j,k$ with $0\le j<k\le n$. A version of it for $n\ge 2$ replaces all $-j$ with $j$ and eliminates all $j=k-1$. Both the GGR conjecture and its version were recently proved by the authors using non-inductive proofs based on highly non-trivial combinatorial algorithms. This article provides a second, inductive, algebraic proof to each of these theorems, based on a reduction to (Laurent) polynomials.

Détail

Reçu le : 2021-06-10
Révisé le : 2022-04-30
Accepté le : 2022-08-24
Publié le : 2023-01-12

DOI : 10.5802/crmath.413

Classification : 26A24, 13F20, 15A03, 26A27

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits