Two Pointwise Characterizations of the Peano Derivative

TL;DR

This paper characterizes the Peano derivative using generalized Riemann derivatives, providing new equivalences and a combinatorial proof that advances understanding of generalized differentiation.

Contribution

It introduces two novel characterizations of the Peano derivative in terms of generalized Riemann derivatives, linking classical and generalized differentiation theories.

Findings

First characterization: existence of all generalized Riemann derivatives up to order n is equivalent to the Peano derivative.

Second characterization: Peano derivative exists iff certain shifted derivatives and Gaussian elimination are also present.

Proves a conjecture related to backward shifts, extending previous results on generalized derivatives.

Abstract

We provide the first two examples of sets of generalized Riemann derivatives of orders up to $n$, $n\geq 2$, whose simultaneous existence for all functions~$f$ at~$x$ is equivalent to the existence of the $n$-th Peano derivative $f_{(n)}(x)$. In this way, we begin to understand how the theory of Peano derivatives can be explained exclusively in terms of generalized Riemann derivatives, a bold new principle in generalized differentiation. In 1936, J. Marcinkiewicz and A. Zygmund showed that the existence of $f_{(n)}(x)$ is equivalent to the existence of both $f_{(n-1)}(x)$ and the $n$th generalized Riemann derivative $\widetilde{D}_nf(x)$, based at $x,x+h,x+2h,x+2^2h,\ldots ,x+2^{n-1}h$. Our first characterization of $f_{(n)}(x)$ is that its existence is equivalent to the simultaneous existence of $\widetilde{D}_1f(x),\ldots,\widetilde{D}_nf(x)$. Our second characterization is that the existence of $f_{(n)}(x)$ is equivalent to the existence of $\widetilde{D}_1f(x)$ and of all $n(n-1)/2$ forward shifts, \[ D_{k,j}f(x)=\lim_{h\rightarrow 0} h^{-k}\sum_{i=0}^k(-1)^i\binom ki f(x+(k+j-i)h), \] for $j=0,1,\ldots,k-2$, of the $k$-th Riemann derivatives $D_{k,0}f(x)$, for $k=2,\ldots ,n$. The proof of the second result involves an interesting combinatorial algorithm that starts with consecutive forward shifts of an arithmetic progression and yields a geometric progression, using two set-operations: dilation and combinatorial Gaussian elimination. This result proves a variant of a 1998 conjecture by Ginchev, Guerragio and Rocca, predicting the same outcome for backward shifts instead of forward shifts. The conjecture has been recently settled in [5], with a proof that has this variant's proof as a prerequisite.