The Regular Convergence of Multiple Series

Abstract

Denote Z<Stack><Subscript>+</Subscript><Superscript>d</Superscript></Stack>the set of d — tuples ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}} = \left( {{k_1}, \ldots, {k_d}} \right)$ with positive integers for coordinates. A d-multiple series $\sum {{u_{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}}}}} = \sum {\left\{ {{u_{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}}}}}:{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}} \in Z_{ + }^d} \right\}} }$, where the summation is extended over ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}} \in Z_{ + }^d$, is said to converge regularly if for every positive η there exists a number N = N(η) so that ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}} \in Z_{ + }^d$ for every rectangle R = $R = \left\{ {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}} \in Z_{ + }^d:{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ell }} \leqslant {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}} \leqslant {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{m}}} \right\}$ provided max(l₁,…, l_d) > N and ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{m}} \geqslant {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ell }}$. Convergence in Pringsheim’s sense follows from regular convergence, but the converse implication is not true in case d &#x2A7E 2. A benefit of the notion of regular convergence is that it makes possible to extend the validity of Kronecker’s lemmas from single series to multiple series and these extensions meet a number of applications, among others, in the theory of multiple orthogonal series and of random fields.