On existence of a universal function for L p [0,1] with p ∈(0,1)

We show that, for every number p a (0, 1), there is g a L (1)[0, 1] (a universal function) that has monotone coefficients c (k) (g) and the Fourier-Walsh series convergent to g (in the norm of L (1)[0, 1]) such that, for every f a L (p) [0, 1], there are numbers delta (k) = +/- 1, 0 and an increasing sequence of positive integers N (q) such that the series a (k=0) (+a) delta (k) c (k) (g)W (k) (with {W (k) } theWalsh system) and the subsequence , alpha a (-1, 0), of its Cesaro means converge to f in the metric of L (p) [0, 1].