A characterization of the Peano derivative

Abstract:

For each choice of parameters $\{ {a_i},{b_i}\} ,i = 0,1, \ldots ,n + e$, satisfying certain simple conditions, the expression \[ \lim \limits _{h \to 0} {h^{ - n}}\sum \limits _{i = 0}^{n + e} {{a_i}f(x + {b_i}h)} \] yields a generalized nth derivative. A function f has an nth Peano derivative at x if and only if all the members of a certain subfamily of these nth derivatives exist at x. The result holds for the corresponding ${L^p}$ derivatives. A uniformity lemma in the proof (Lemma 2) may be of independent interest. Also, a new generalized second derivative is introduced which differentiates more functions than the ordinary second derivative but fewer than the second Peano derivative.