On the divergence of the Fejér means of integrable functions on two-dimensional Vilenkin groups

Summary

In 1992, M&oacute;ricz, Schipp and Wade [6] proved the a.e. convergence of the double (<Emphasis Type=”Italic”>C</Emphasis>,1) means of the Walsh--Fourier series &sgr;<Emphasis Type=”Italic”><Subscript>n </Subscript>f </Emphasis> &rarr; <Emphasis Type=”Italic”>f</Emphasis> (min (<Emphasis Type=”Italic”>n</Emphasis><Subscript>1</Subscript>, <Emphasis Type=”Italic”>n</Emphasis><Subscript>2</Subscript>)&rarr;&infin;, <Emphasis Type=”Italic”>n</Emphasis>=(<Emphasis Type=”Italic”>n</Emphasis><Subscript>1</Subscript>,<Emphasis Type=”Italic”>n</Emphasis><Subscript>2</Subscript>) &isin; <Emphasis Type=”Bold”>N</Emphasis><Superscript>2</Superscript>) for functions in <Emphasis Type=”Italic”>L </Emphasis>log<Superscript>+</Superscript> <Emphasis Type=”Italic”>L</Emphasis> ([0,1)<Superscript>2</Superscript>). This result for bounded Vilenkin groups is generalized by Weisz [10]. We show that these results can not be improved with respect to two-dimensional bounded Vilenkin groups (not only the two-dimensional Walsh group). We prove that for all measurable functions &dgr; : [0,+&infin;) &rarr; [0,+&infin;), lim <Subscript>t &rarr; &infin;</Subscript> &dgr;(<Emphasis Type=”Italic”>t</Emphasis>) = 0, <Emphasis Type=”Italic”>G<Subscript>m</Subscript></Emphasis>x<Emphasis Type=”Italic”>G<Subscript>m&apos;</Subscript></Emphasis> two-dimensional bounded Vilenkin group we have an f &isin; <Emphasis Type=”Italic”>L </Emphasis>log<Superscript>+</Superscript> <Emphasis Type=”Italic”>L</Emphasis>&dgr;(<Emphasis Type=”Italic”>L</Emphasis>) such that &sgr;<Emphasis Type=”Italic”><Subscript>n</Subscript>f</Emphasis> does not converge to <Emphasis Type=”Italic”>f</Emphasis> a.e. (in the Pringsheim sense).