The classification of generalized Riemann derivatives

Abstract:

A generalized $n$th Riemann derivative of a real function $f$ at $x$ is given by \[ \lim _{h\rightarrow 0}\frac 1{h^n} \sum _{i=1}^{m}A_{i}f(x+a_{i}h). \] The above sum $\Delta _{\mathcal {A}}$ is called an $n$th generalized Riemann difference. The data vector $\mathcal {A}=\{A_1,\ldots ,A_m;a_1,\ldots ,a_m\}$ satisfies suitable conditions that make the limit agree with $f^{(n)}(x)$ whenever this exists. We explain the underlying reason for a surprising relationship between certain generalized $n$th Riemann derivatives recently discovered by Ash, Catoiu, and Csörnyei. We characterize all pairs $(\Delta _{\mathcal {A}},\Delta _{\mathcal {B}})$ of generalized Riemann differences of any orders for which $\mathcal {A}$-differentiability implies $\mathcal {B}$-differentiability. Two generalized Riemann derivatives $\mathcal {A}$ and $\mathcal {B}$ are equivalent if a function has a derivative in the sense of $\mathcal {A}$ at a real number $x$ if and only if it has a derivative in the sense of $\mathcal {B}$ at $x$. We determine the equivalence classes for this equivalence relation. The classification of these by now classical objects of real analysis was made possible by using a less known and less studied notion from algebra, the group algebra of the multiplicative group $\mathbb {R}^{+}$ of the positive reals over the field $\mathbb {R}$ of real numbers.