论文标题
高斯里曼衍生物
Gaussian Riemann Derivatives
论文作者
论文摘要
J. Marcinkiewicz和A. Zygmund在1936年证明，对于所有功能$ f $和$ x $，存在$ n $ n $ th peano衍生品$ f _ {（n）}（x）$等于存在$ f _ {（n-1）} $ f _ {（n-1）} $和$ n $ n $ n $ n $ n $ n $ n $ n $ $ \ widetilde {d} _nf（x）$，基于$ x，x+h，x+2h，x+2^2h，\ ldots，x+2^{n-1} h $。 For $q\neq 0,\pm 1$, we introduce: two $q$-analogues of the $n$-th Riemann derivative ${D}_nf(x)$ of~$f$ at~$x$, the $n$-th Gaussian Riemann derivatives ${_q}{D}_nf(x)$ and ${_q}{\bar d} _nf（x）$是$ n $ th概括的riemann衍生物，基于$ x，x+h，x+qh，x+q^2H，\ ldots，x+q^{n-1}和$ n $ -th对称的Riemann衍生品$ {d} _n^sf（x）$，$ n $ -th对称的高斯riemann衍生$ {_ q} {d} {d} {d} _n^sf（x）$ qh，x \ pm q^2H，\ ldots，x \ pm q^{m-1} h $，其中$ m = \ lfloor（n+1）/2 \ rfloor $ and〜 $（x）$表示$ x $仅用于$ n $偶数。我们为高斯二项式系数而言，为它们相关的差异提供了确切的表达。我们表明，两个$ n $ th th th th dh $ n $ th $ n $ th对称的高斯导数满足了定理的对称版本；我们推测，这两个结果对于每个较大类别的广义Riemann衍生物都是错误的，从而扩展了Ash和Catoiu最近的两个猜想，这两种猜想都是通过在少数情况下回答的。
J. Marcinkiewicz and A. Zygmund proved in 1936 that, for all functions $f$ and points $x$, the existence of the $n$th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of both $f_{(n-1)}(x)$ and the $n$th generalized Riemann derivative $\widetilde{D}_nf(x)$, based at $x,x+h,x+2h,x+2^2h,\ldots ,x+2^{n-1}h$. For $q\neq 0,\pm 1$, we introduce: two $q$-analogues of the $n$-th Riemann derivative ${D}_nf(x)$ of~$f$ at~$x$, the $n$-th Gaussian Riemann derivatives ${_q}{D}_nf(x)$ and ${_q}{\bar D}_nf(x)$ are the $n$-th generalized Riemann derivatives based at $x,x+h,x+qh,x+q^2h,\ldots ,x+q^{n-1}h$ and $x+h,x+qh,x+q^2h,\ldots ,x+q^{n}h$; and one analog of the $n$-th symmetric Riemann derivative ${D}_n^sf(x)$, the $n$-th symmetric Gaussian Riemann derivative ${_q}{D}_n^sf(x)$ is the $n$-th generalized Riemann derivative based at $(x),x\pm h,x\pm qh,x\pm q^2h,\ldots ,x\pm q^{m-1}h$, where $m=\lfloor (n+1)/2\rfloor $ and~$(x)$ means that $x$ is taken only for $n$ even. We provide the exact expressions for their associated differences in terms of Gaussian binomial coefficients; we show that the two $n$th Gaussian derivatives satisfy the above classical theorem, and that the $n$th symmetric Gaussian derivative satisfies a symmetric version of the theorem; and we conjecture that these two results are false for every larger classes of generalized Riemann derivatives, thereby extending two recent conjectures by Ash and Catoiu, both of which we update by answering them in a few cases.
