On Axerʼs theorem

In Axerʼs theorem, conditions on a function of bounded variation and an infinite series of real numbers imply an order condition on the multiplicative convolution of the series and the function. We prove an extension of Axerʼs theorem. We show that although the converse of Axerʼs theorem is false without further restriction on the function, with such a restriction one can prove the converse of the strengthened theorem. We also consider the changes in Axerʼs theorem resulting from assuming the function to be in ΛBV or ΦBV.