论文标题
Peano衍生物的两个点特征
Two Pointwise Characterizations of the Peano Derivative
论文作者
论文摘要
我们提供了前两个示例的订单，最高$ n $，$ n \ geq 2 $的订单的普通衍生物，其同时存在所有功能的〜$ f $ at〜$ x $等于存在$ n $ n $ th peano derivative deano derivativative $ f _ {（n）}（x）$。通过这种方式，我们开始理解如何专门用广义的Riemann衍生物来解释Peano衍生物的理论，这是一种大胆的广义分化新原则。 1936年，J。Marcinkiewicz和A. Zygmund表明，存在$ f _ {（n）}（x）$的存在与存在$ f _ {（n-1）}（x）$和$ n $ th $ n $ th th riemann derivative $ \ \ \ \ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\图$ x，x+h，x+2h，x+2^2H，\ ldots，x+2^{n-1} h $。我们对$ f _ {（n）}（x）$的第一个表征是，它的存在等效于同时存在$ \ widetilde {d} _1f（x），\ ldots，\ widetilde {d} _nf（x）$。我们的第二个特征是，$ f _ {（n）}（x）$的存在相当于存在$ \ widetilde {d} _1f（x）$，以及所有$ n（n-1）/2 $ forthly shifts，\ [d_ {d _ {k，j，j} f（x） h^{ - k} \ sum_ {i = 0}^k（-1）^i \ binom ki f（x+（x+（k+j-i）h），\ \]，\ j = 0,1，\ ldots，k-2 $，$ k $ - $ k $ -k $ -k $ - ththriemann demann demann d_ $ d_ { 第二个结果的证明涉及一种有趣的组合算法，该算法从算术进展的连续向前移动开始，并使用两种固定操作：扩张和组合高斯消除。该结果证明了Ginchev，Guerragio和Rocca的1998年猜想的一种变体，预测了向后移动而不是向前移动的相同结果。该猜想最近在[5]中解决了，证明了该变体的证明是先决条件。
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.
