A new proof of the GGR conjecture

Ash, J. Marshall; Catoiu, Stefan; Fejzić, Hajrudin

doi:10.5802/crmath.413

Analyse fonctionnelle

A new proof of the GGR conjecture

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

¹Department of Mathematics, DePaul University, Chicago, IL 60614
²Department 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 $f_{(n)} (x)$ 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 :

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

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     title = {A new proof of the {GGR} conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {349--353},
     year = {2023},
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Ash, J. Marshall; Catoiu, Stefan; Fejzić, Hajrudin. A new proof of the GGR conjecture. Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 349-353. doi: 10.5802/crmath.413

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

[12] de la Vallée Poussin, Charles-Jean 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

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