  We prove that the class of Muckenhoupt A_p weights coincides with the
intersection of finitely many suitable translates of dyadic A_p, in both the
one-parameter and multiparameter cases, and that the analogous results hold for
the reverse H\"older class RH_p, for doubling measures, and for the space VMO
of functions of vanishing mean oscillation. We extend to the multiparameter
(product) space BMO of functions of bounded mean oscillation the corresponding
one-parameter BMO result due to T. Mei, by means of the Carleson-measure
characterization of multiparameter BMO. Our results hold in both the compact
and non-compact cases. In addition, we survey several definitions of VMO and
prove their equivalences, in the continuous, dyadic, one-parameter and
multiparameter cases. We show that the weighted Hardy space H^1(\omega) is the
sum of finitely many suitable translates of dyadic weighted H^1(\omega), and
that the weighted maximal function is pointwise comparable to the sum of
finitely many dyadic weighted maximal functions for suitable translates of the
dyadic grid and for each doubling weight \omega.
