Functions that have no first order derivative might have fractional derivatives of all orders less than one
type: Item
instance of: scholarly article
title: Functions that have no first order derivative might have fractional derivatives of all orders less than one
zbMATH Open document ID: 0820.26002
author: Bertram Ross
author: E. R. Love
author: S. G. Samko
published in: Real Analysis Exchange
publication date: 1995-09-10T00:00:00Z
review text: The main question studied in the paper is whether there exists a function \(f(x)\), \(x\in \mathbb{R}^ 1\), which everywhere has a continuous fractional derivative \(D^ \alpha\) of any order \(\alpha< \alpha_ 0\) but nowhere has the fractional derivative just of order \(\alpha_ 0\). The answer is positive and it is shown that the well-known Weierstrass function \(\sum^ \infty_{n= 0} q^{-\alpha_ 0 n} e^{iq^ n x}\), \(q> 1\), \(\alpha_ 0> 0\), may be used as such an example. The Riemann function \(\sum^ \infty_{n= 1} {\cos n^ 2 x\over n^ 2}\) is also studied from this point of view.
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 26A27
zbMATH DE Number: 756659
zbMATH Keywords: fractional derivative
zbMATH Keywords: Weierstrass function
zbMATH Keywords: Riemann function
reviewed by: S. G. Samko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 756659