论文标题
对高斯与MZ衍生物的反例
Counterexamples to the Gaussian vs. MZ derivatives Conjecture
论文作者
论文摘要
J. Marcinkiewicz和A. Zygmund在1936年证明了特殊的$ n $ th概括的riemann衍生品$ {_ 2} d_nf（x）d_nf（x）$，nodes $ 0,1,2,2^2，\ ldots，\ ldots，2^{n-1} $，对2^{n-1} $，对$ n $ n $ n $ - $ fivity $ fivitivativative forivitivative forivativative forivativative所有$ n-1 $ times peano可区分功能$ f $ at〜 $ x $。将每$ n $ th $ n $ th概括为此属性为MZ导数。最近的纸灰，catoiu和fejzić[以色列J. Math。 {255}（2023）：177--199]引入了$ n $ -th高斯导数，作为$ n $ th的$ n $ th，带有节点$ 0,1，q，q^2，q^2，q^2，q^{n-1} $或$ 1，q，q，q^2，q^2，q^2，q^2，q^2，q^ne q^us q^pm^neq^pm^neq^pm^n wer 1 $，证明高斯衍生物是MZ衍生物，并猜想这些是\ emph {ash ash {ash asl} mz衍生物。在本文中，我们通过两个反例使这一猜想无效。呈现的顺序允许在每个反例子之后更新猜想。第一个反例的证明是简单的，通过广义riemann衍生物的尺度。第二个的证明涉及将灰，catoiu和Chin的普遍的Riemann衍生物分类[Proc。阿米尔。数学。 SOC {146}（2018）：3847--3862]。所有结果的对称版本也包括在内。
J. Marcinkiewicz and A. Zygmund proved in 1936 that the special $n$-th generalized Riemann derivative ${_2}D_nf(x)$ with nodes $0,1,2,2^2,\ldots, 2^{n-1}$, is equivalent to the $n$-th Peano derivative $f_{(n)}(x)$, for all $n-1$ times Peano differentiable functions $f$ at~$x$. Call every $n$-th generalized Riemann derivative with this property an MZ derivative. The recent paper Ash, Catoiu, and Fejzić [Israel J. Math. {255} (2023):177--199] introduced the $n$-th Gaussian derivatives as the $n$-th generalized Riemann derivatives with nodes either $0,1,q,q^2,\ldots ,q^{n-1}$ or $1,q,q^2,\ldots ,q^{n}$, where~$q\neq0,\pm 1$, proved that the Gaussian derivatives are MZ derivatives, and conjectured that these are \emph{all} MZ derivatives. In this article, we invalidate this conjecture by means of two counterexamples. The order in which these are presented allows an update of the conjecture after each counterexample. The proof of the first counterexample is simple, by scales of generalized Riemann derivatives. The proof of the second involves the classification of generalized Riemann derivatives of Ash, Catoiu, and Chin [Proc. Amer. Math. Soc {146} (2018):3847--3862]. Symmetric versions of all the results are also~included.
