Harmonic and logarithmic summability of orthogonal series are equivalent up to a set of measure zero

摘要

We prove Tauberian theorems from J_p-summability methods of power series type to ordinary convergence, respectively M_p-summability methods of weighted means. Particular cases are the Abel and Cesaro, as well as logarithmic and harmonic summability. Besides numerical series, we also consider orthogonal series with coefficients from l_2. In the latter case, it turns out that one of our Tauberian conditions is satisfied almost everywhere on the underlying measure space, thereby proving the claim stated in the title.