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A Generalized Measure Concentrated on a Closed Set of Measure Zero

Authors:

K Yoneda

Institution:

(1) Department of Mathematics and Information Sciences, Osaka Prefecture University Sakai, Osaka, Japan

Abstract:

When E is a closed set of measure zero in the dyadic group and the Walsh series $\sum\nolimits_{k = 0}^\infty {\hat \mu (k)\omega _k (x)}$ satisfies

$\mathop {\lim }\limits_{n \to 0} \sum\limits_{k = 0}^{2^n - 1} {\hat \mu (k)\omega _k (x)} = 0 everywhere except on E and \hat \mu (0) \ne 0,$

, then for some c > 0,

$mes\left( {x : \mathop {sup}\limits_n \left| {\sum\limits_{k = 0}^{2^n - 1} {\hat \mu (k)\omega _k (x)} } \right| > t} \right) > \frac{c}{t} for sufficiently large t.$

Consequently any control function of the a.e. convergence is not L/(log⁺ L)-integrable.

Keywords:

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