GGR猜想的推广

摘要：

对于每个正整数$n$、函数$f$和点$c$，GGR定理指出$f$在$c$上是$n$乘Peano可微的当且仅当$f$在$c$上是$n-1$乘Peano可微的，并且$f$在$c$上的以下$n$阶广义黎曼导数存在：对于$k=0,\ldots,n-1$，存在\[ \lim_{h\rightarrow 0}\frac 1{h^{n}}\sum_{i=0}^n(-1)^i\binom{n}{i}f(c+(n-i-k)h), \]。该定理最近在[AC 2]中得到证明，自1998年以来一直是Ghinchev，Guerragio和Rocca的猜想。我们提供了一个新的证明这一定理的基础上，它产生了许多新的$n$ -th黎曼光滑条件，可以发挥上述一套GGR定理的作用。

英文摘要：

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.

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