The main results of the paper are the following: the Fourier expansion of $f \in L_p$, $1 \lt p \lt \infty$, with respect to the Walsh transform of a wavelet type system converges a.e. to $f$ and if $f \in L_1$ then the same is true for the Cesàro means.
"The Walsh transform of wavelet type systems: convergence almost everywhere." Funct. Approx. Comment. Math. 32 79 - 98, 2004. https://doi.org/10.7169/facm/1538186628