Real Analysis Exchange

Extreme Results on Certain Generalized Riemann Derivatives

John C. Georgiou

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Abstract

In this paper the following question is investigated. Given a natural number \(r\) and numbers \(\alpha_j,\beta_j\) for \(j=0,1,\dots,r\) satisfying \( \alpha_0 <\alpha_1 < \dots \lt \alpha_r \) and \begin{equation*} \sum_{j=0}^{r} \beta_j \alpha_j^k= \begin{cases} 0 & \text{if \(k=0,1,\dots,r-1\)}\\ r!& \text{if \(k=r\) } \end{cases} , \end{equation*} is there a \( 2\pi\)-periodic, \( r-1\) times continuously differentiable function \( f\) such that \begin{equation*} \limsup_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \limsup_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \infty, \end{equation*} \begin{equation*} \liminf_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \liminf_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = - \infty \end{equation*} for every \( x \in \mathbb{R} \)?

Article information

Source
Real Anal. Exchange, Volume 40, Number 1 (2015), 193-208.

Dates
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.rae/1435759203

Mathematical Reviews number (MathSciNet)
MR3365398

Zentralblatt MATH identifier
06848831

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 26A51: Convexity, generalizations 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26B08
Secondary: 40A30: Convergence and divergence of series and sequences of functions 54C50: Special sets defined by functions [See also 26A21] 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]

Keywords
non-differentiability convexity divided differences generalized derivatives

Citation

Georgiou, John C. Extreme Results on Certain Generalized Riemann Derivatives. Real Anal. Exchange 40 (2015), no. 1, 193--208. https://projecteuclid.org/euclid.rae/1435759203


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