A new proof of the GGR conjecture

J. Marshall Ash; Stefan Catoiu; Hajrudin Fejzić

doi:10.5802/crmath.413

Analyse fonctionnelle

A new proof of the GGR conjecture

J. Marshall Ash ¹ ; Stefan Catoiu ¹ ; Hajrudin Fejzić ²

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

Comptes Rendus. Mathématique, Volume 361 (2023), 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 $f_{(n)} (x)$ is equivalent to the existence of all $n (n + 1) / 2$ generalized Riemann derivatives,

D_{k, - j} f (x) = lim_{h \to 0} \frac{1}{h^{n}} \sum_{i = 0}^{k} {(- 1)}^{i} (\binom{k}{i}) f (x + (k - i - j) h),

for $j, k$ with $0 \leq j < k \leq n$ . A version of it for $n \geq 2$ 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.

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

Affiliations des auteurs :

J. Marshall Ash ¹ ; Stefan Catoiu ¹ ; Hajrudin Fejzić ²

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G1_349_0,
     author = {J. Marshall Ash and Stefan Catoiu and Hajrudin Fejzi\'c},
     title = {A new proof of the {GGR} conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {349--353},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.413},
     language = {en},
}

TY  - JOUR
AU  - J. Marshall Ash
AU  - Stefan Catoiu
AU  - Hajrudin Fejzić
TI  - A new proof of the GGR conjecture
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 349
EP  - 353
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.413
LA  - en
ID  - CRMATH_2023__361_G1_349_0
ER  -

%0 Journal Article
%A J. Marshall Ash
%A Stefan Catoiu
%A Hajrudin Fejzić
%T A new proof of the GGR conjecture
%J Comptes Rendus. Mathématique
%D 2023
%P 349-353
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.413
%G en
%F CRMATH_2023__361_G1_349_0

J. Marshall Ash; Stefan Catoiu; Hajrudin Fejzić. A new proof of the GGR conjecture. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 349-353. doi : 10.5802/crmath.413. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.413/

Bibliographie
Cité par

[1] J. M. Ash; Stefan Catoiu Quantum symmetric $L^{p}$ derivatives, Trans. Am. Math. Soc., Volume 360 (2008) no. 2, pp. 959-987 | DOI | MR | Zbl

[2] J. Marshall Ash; Stefan Catoiu Characterizing Peano and symmetric derivatives and the GGR conjecture’s solution, Int. Math. Res. Not., IMRN (2022) no. 10, pp. 7893-7921 | DOI | MR | Zbl

[3] J. Marshall Ash; Stefan Catoiu; Hajrudin Fejzić Gaussian Riemann derivatives (2022) (https://arxiv.org/abs/2211.09209, to appear in Israel J. Math., 23 pp., online first)

[4] J. Marshall Ash; Stefan Catoiu; Hajrudin Fejzić Two pointwise characterizations of the Peano derivative (2022) (https://arxiv.org/abs/2209.04088, preprint)

[5] J. Marshall Ash; Stefan Catoiu; Ricardo Ríos-Collantes-de-Terán On the n $^{th}$ quantum derivative, J. Lond. Math. Soc., Volume 66 (2002) no. 1, pp. 114-130 | MR | Zbl

[6] Stefan Catoiu A differentiability criterion for continuous functions, Monatsh. Math., Volume 197 (2022) no. 2, pp. 285-291 | DOI | MR | Zbl

[7] Ivan Ginchev; Angelo Guerraggio; Matteo Rocca Equivalence of Peano and Riemann derivatives, Generalized convexity and optimization for economic and financial decisions (Verona, 1998), Pitagora Editrice, Bologna, 1998, pp. 169-178 | Zbl

[8] Ivan Ginchev; Matteo Rocca On Peano and Riemann derivatives, Rend. Circ. Mat. Palermo, Volume 49 (2000) no. 3, pp. 463-480 | DOI | MR | Zbl

[9] Józef Marcinkiewicz; Antoni Zygmund On the differentiability of functions and summability of trigonometric series, Fundam. Math., Volume 26 (1936), pp. 1-43 | DOI | Zbl

[10] Giuseppe Peano Sulla formula di Taylor, Atti Acad. Sci. Torino, Volume 27 (1892), pp. 40-46

[11] Bernhard Riemann Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe, Ges. Werke, 2. Aufl., Leipzig, Dieterichschen Buchhandlung, 1892 (1867), pp. 227-271

[12] Charles-Jean de la Vallée Poussin Sur l’approximation des fonctions d’une variable réelle et de leurs dérivées par les pôlynomes et les suites limitées de Fourier, Belg. Bull. Sciences, Volume 1908 (1908), pp. 193-254 | Zbl

Cité par Sources :

Commentaires - Politique