Geodesic
On Walsh series with monotone coefficients
Izvestiya. Mathematics , Tome 63 (1999) no. 1, pp. 37-55.

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We prove that if $a_n\downarrow 0$ and $\sum_{n=0}^\infty a_n^2=+\infty$ then the Walsh series $\sum_{n=0}^\infty a_nW_n(x)$ has the following property. For any measurable function $f(x)$ which is finite almost everywhere, there are numbers $\delta_n=0,\pm 1$ such that the series $\sum_{n=0}^\infty\delta_na_nW_n(x)$ converges to $f(x)$ almost everywhere. This assertion complements and strengthens previously known results about universal Walsh series and Walsh null-series. 
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G. G. Gevorkyan; K. A. Navasardyan. On Walsh series with monotone coefficients. Izvestiya. Mathematics , Tome 63 (1999) no. 1, pp. 37-55. http://geodesic.mathdoc.fr/item/IM2_1999_63_1_a1/

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