Greedy approximation with respect to certain subsystems of the Walsh orthonormal system

In an article that appeared in 1967, J.J. Price has shown that there is a vast family of subsystems of the Walsh orthonormal system each of which is complete on sets of large measure. In the present work it is shown that the greedy algorithm, when applied to functions in $L^{1}[0,1]$, is surprisingly effective for these nearly-complete families. Indeed, if $\Phi$ is such a subsystem of the Walsh system, then to each positive $\varepsilon$, however small, there corresponds a Lebesgue measurable set $E$ such that for every $f$, Lebesgue integrable on $[0,1]$, the greedy approximants to $f$, associated with $\Phi$, converge, in the $L^{1}$ norm, to an integrable function $g$ that coincides with $f$ on $E$.