On Peano and Riemann derivatives
type: Item
instance of: scholarly article
title: On Peano and Riemann derivatives
zbMATH Open document ID: 1010.26005
DOI: 10.1007/BF02904258
author: Ivan Ginchev
author: Matteo Rocca
publication date: 2003-03-18T00:00:00Z
review text: For a positive integer \(n\) denote by \(C_n\) the following statement: Let \(f\) be a real function defined in a neighborhood of a \(t_0\in\mathbb R\). Then the \(n\)-th order Peano derivative \(D^{(n)}_Pf(t_0)\) exists if and only if the Riemann derivative \(D^{(k)}_Rf(t_0)\) exists for \(k=0,1,\dots ,n\). In this case, \(D^{(n)}_Rf(t_0)=D^{(n)}_Pf(t_0)\). The authors conjecture that \(C_n\) is true for any positive integer \(n\). Actually, the implication Peano \(\Rightarrow\) Riemann is easy, the opposite one is a problem. In the paper, the authors formulate an equivalent conjecture involving a functional determinant of order \(2n+1\) and, using it, they check \(C_n\) for \(n\leq 8\).
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 1884295
zbMATH Keywords: Riemann derivative
zbMATH Keywords: Peano derivative
reviewed by: Jaroslav Smítal
MaRDI profile type: Publication
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1884295