On the Weyl Multipliers for General Haar and Franklin Systems

Abstract

In the work the almost everywhere (a.e.) convergence (absolute convergence) of series by the general Haar and Franklin systems corresponding to weakly regular division of the segment \([0,1]\) are compared. It is proved that if a series by the general Haar system diverges (absolutely diverges) on a set \(E\), then the series by the general Franklin system with the same coefficients diverges (absolutely diverges) a.e. in \(E\). As a consequence, it is obtained that if a sequence \(\omega_{n}\) is not a Weyl multiplier for unconditional a.e. convergence of series by the general Haar system, then it is not a Weyl multiplier for unconditional a.e. convergence of series by the general Franklin series.