An example of a zero series expansion in the Walsh system

We construct an example of a zero series expansion in the Walsh system which converges to zero outside some closed $M$ set of zero measure and converges to $+\infty$ at each point of this set. This shows, in particular, that in the theorem which says that a Walsh series which converges everywhere to a finite symmetric function is a Fourier series it is impossible to omit the requirement of finiteness and allow convergence of the series on a set of zero measure to an infinity of specified sign.