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

We prove Tauberian theorems from J(p)-summability methods of power series t ype to ordinary convergence, respectively M-p-summability methods of weight ed means. Particular cases are the Abel and Cesaro. as well as logarithmic and harmonic summability. Besides numerical series, we also consider orthog onal series with coefficients from l(2). In the latter case, it turns out t hat one of our Tauberian conditions is satisfied almost everywhere on the u nderlying measure space, thereby proving the claim stated in the title.