For every orthonormal system of functions \₍\ on (0, \, 1), pointwise bounded by a function f L^2 (0, \, 1), that is, |₍ (x) | f (x), it is possible to construct an orthonormal system \₍\ with |₍ (x) | 1, such that the series a₍₍ and a₍₍ converge almost everywhere simultaneously, for every \a₍\^2.
V. G. Dilanyan (Sun,) studied this question.