  The main aim of this paper is to investigate the sequences of positive
numbers, for which multiplication with Fourier coefficients of functions $f\in$
Lip1 class provides absolute convergence of Fourier series. In particular we
found special conditions for the functions of orthonormal system, for which the
above sequences are absolute convergence factors of Fourier series of functions
of Lip1 class. It is established that the resulting conditions are best
possible in certain sense.
