  We prove a sufficient condition for frame-type wavelet series in $L^p$, the
Hardy space $H^1$, and BMO. For example, functions in these spaces are shown to
have expansions in terms of the Mexican hat wavelet, thus giving a strong
answer to an old question of Meyer.
  Bijectivity of the wavelet frame operator acting on Hardy space is
established with the help of new frequency-domain estimates on the
Calder\'on-Zygmund constants of the frame kernel.
