  S. Banach \cite{Banach} proved that good differential properties of function
do not guarantee the a.e. convergence of the Fourier series of this function
with respect to general orthonormal systems (ONS). On the other hand it is very
well known that a sufficient condition for the a.e. convergence of an
orthonormal series is given by the Menshov-Rademacher Theorem.
  The paper deals with sequence of positive numbers $(d_n)$ such that
multiplying the Fourier coefficients $(C_n(f))$ of functions with bounded
variation by these numbers one obtains a.e. convergent series of the form
$\sum_{n=1}^{\infty}d_n C_n(f) \varphi_n (x).$ It is established that the
resulting conditions are best possible.
