Multidimensional Riemann derivatives (Q2833670)

In this paper, the authors consider three kinds of differentiability or order~\(n\) for functions of several real variables. These are: Peano differentiability, Riemann differentiability and symmetric Riemann differentiability. It is easy to prove that the existence of the Peano differential implies pointwise the existence of both the Riemann and the symmetric Riemann differential. The main result is that these three generalized differentials exist simultaneously at almost every points.NEWLINENEWLINEThe authors also consider the corresponding Lipschitz conditions and obtain results of Rademacher type.NEWLINENEWLINELet \(f: \mathbb{R}^{d}\to\mathbb{R}\) be a measurable function. A polynomial approximation for~\(f\) of degree~\(n\) near the point \(x\in\mathbb{R}^{d}\) can be written as NEWLINE\[NEWLINE f(x+h)=\sum^{n}_{k=0} \frac{P_{nk}(x,h)}{k!} +o(\|h\|^{n}), NEWLINE\]NEWLINE where \(P_{nk}\) is a homogeneous polynomial of degree~\(k\) in the variable~\(h\) with coefficients depending on~\(x\). We will write NEWLINE\[NEWLINE P_{nk}(x,h)=\sum_{|\kappa|=k}\frac{k!}{\kappa!}f_{\kappa}(x) h^{\kappa} NEWLINE\]NEWLINE where \(\kappa=(\kappa_{1},\dots,\kappa_{d})\in Z^{d}\), \(h^{\kappa} =h_{1}^{\kappa_{1}},\dots, h_{d}^{\kappa_{d}}\) and \(|\kappa|=\kappa_{1},\dots,\kappa_{d}\). This defines \(f_{\kappa}(x)\) as the \(\kappa\)th partial Peano derivative of~\(f\) at~\(x\).NEWLINENEWLINEThen, \(f\) has an \(n\)th \textit{Peano derivative} at~\(x\) if \(f(x+h)=\sum\limits_{|\kappa|\leq h}f_{\kappa}(x) h^{\kappa} /\kappa! +o(\|h\|^{n})\).NEWLINENEWLINEAlso, the corresponding \textit{Lipschitz condition} of order~\(n\) at~\(x\) is obtained; this is written as \(f(x+h)=\sum\limits_{|\kappa|\leq n-1} f_{\kappa}(x) h^{\kappa}|_{\kappa!}+o(\|h\|^{n})\).NEWLINENEWLINEWe introduce now the other two types of differentiability. \(f\) is \textit{Riemann differentiable} of order~\(n\) at~\(x\) if NEWLINE\[NEWLINE \sum^{n}_{i=0} (-1)^{n-i} \binom{n}{i} f(x+ih) =\sum_{|\kappa|=n}\frac{n!}{\kappa!} f_{\kappa}(x) h^{\kappa}+o(\|h\|^{n}) NEWLINE\]NEWLINE and the corresponding \textit{Riemann-Lipschitz condition} of order~\(n\) is NEWLINE\[NEWLINE \sum^{n}_{i=0}(-1)^{n-i}\binom{n}{i} f(x+ih)=o(\|h\|^{n}). NEWLINE\]NEWLINE If, in the last two definitions, NEWLINE\[NEWLINE \sum^{n}_{i=0} (-1)^{n-i} \binom{n}{i} f(x+ih) NEWLINE\]NEWLINE is replaced by NEWLINE\[NEWLINE \sum^{n}_{i=0} (-1)^{n-i} \binom{n}{i} f\left(x+\left(i-\frac{n}{2}\right)h\right), NEWLINE\]NEWLINE the function~\(f\) is respectively called \textit{symmetric Riemann differentiable} of order~\(n\) at~\(x\) and \textit{symmetric Riemann Lipschitz} of order~\(n\) at~\(x\).NEWLINENEWLINEThe main result can be stated now in the following form:NEWLINENEWLINELet \(n\) be a positive integer, and suppose that, for every~\(x\) in a measurable set \(E\subset \mathbb{R}^{d}\), the measurable function~\(f: \mathbb{R}^{d}\to\mathbb{R}\) satisfies one of the following six conditions. {\parindent=0.7cm\begin{itemize}\item[1.] \(f\) is \(n\)th Peano differentiable at~\(x\), \item[2.] \(f\) is Lipschitz of order~\(n\) at~\(x\), \item[3.] \(f\) is Riemann differentiable of order~\(n\) at~\(x\), \item[4.] \(f\) is Riemann Lipschitz of order~\(n\) at~\(x\), \item[5.] \(f\) is symmetric Riemann differentiable of order~\(n\) at~\(x\), \item[6.] \(f\) is symmetric Riemann Lipschitz of order~\(n\) at~\(x\). NEWLINENEWLINE\end{itemize}} Then, \(f\) satisfies all six conditions at almost every point of~\(E\).