The classification of generalized Riemann derivatives

A generalized nth Riemann derivative of a real function f at x is given by \displaystyle \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 nth 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 nth 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.