On traces of series of means of second differences (Q1374968)

Let \(f:{\mathbb{R}}^n\rightarrow {\mathbb{R}}\) be locally integrable on every compact set and \(x:=(x_1, \ldots , x_n)^T\). The trace of \(f\) on the hyperplane \({\mathbb{R}}_{x_n} ^{n-1}\) is defined as a function \(g: {\mathbb{R}}^{n-1}\rightarrow {\mathbb{R}}\) such that there exits an \(f_0: {\mathbb{R}}^n\rightarrow {\mathbb{R}}\) coinciding with \(f\) almost everywhere and \[ \lim_{\triangle x_n\rightarrow 0} \| f_0(\cdot , x_n+\triangle x_n)-g(\cdot)\|_{L_p({\mathbb{P}}_{x_n}^{n-1}(a))}=0 \] is satisfied for any \(a>0\), where \({\mathbb{P}}_{x_n}^{n-1}(a)=\{x: | x_i| \leq a, i=1,\ldots , n\}\cap {\mathbb{R}}_{x_n}^{n-1}\). In this paper a method for finding traces of functions is suggested, which is based on the expansion into series of mean functions and second differences of the given functions.