Rational orthogonal systems in approximating analytic functions have attracted considerable interest. Among which adaptive Fourier decomposition, abbreviated as AFD, was recently established. An AFD is a sparse representation using a Takenaka– Malmquist (TM) system whose parameters are optimally selected according to the given signal. TM systems have been proved to be Schauder systems in the corresponding Banach spaces Hp, 1 < p < ∞. In the present paper, from the methodology point of view we give an alternative definition of the Hardy spaces by using the periodic Lusin area function. We extend the Botchkariev–Meyer–Wojtaszcyk Theorem to rational function systems. By using Meyer’s bimodal wavelet and the Fefferman–Stein vector valued maximum operator we prove that under certain conditions the rational systems become unconditional bases in the Banach space Hp(D), 1 < p < ∞.