Abstract. There are many classical sufficient conditions for the absolute convergence. Convergence of the series $\sum_{j=0}^{\infty }2^jE_2(f;2^j)$, where $E_2(f;2^j)$ is the best approximation in $L^2$ norm of the function $f(x)$ by Walsh--Fourier polynomials of degree not higher than $2^j$, implies the absolute convergence of the Walsh--Fourier series of this function, which is the result of Bernstein and Steckin. We establish a similar result and also give several corollaries of it for the double Walsh--Fourier series. Our results are expressed in terms of the best approximations and Besov spaces.

AMS Subject Classification (1991): 42A20, 42C10

Received September 13, 2005, and in revised form January 24, 2006. (Registered under 5911/2009.)