A Generalization of the GGR Conjecture

For each positive integer \(n\), function \(f\), and point \(c\), the GGR Theorem states that \(f\) is \(n\) times Peano differentiable at \(c\) if and only if \(f\) is \(n-1\) times Peano differentiable at \(c\) and the following \(n\)-th generalized Riemann~derivatives of \(f\) at \(c\) exist: \[ \lim_{h\rightarrow 0}\frac 1{h^{n}}\sum_{i=0}^n(-1)^i\binom{n}{i}f(c+(n-i-k)h), \] for \(k=0,\ldots,n-1\). The theorem has been recently proved in [AC2] and has been a conjecture by Ghinchev, Guerragio, and Rocca since 1998. We provide a new proof of this theorem, based on a generalization of it that produces numerous new sets of \(n\)-th Riemann smoothness conditions that can play the role of the above set in the GGR Theorem.