  In this paper, we develop a theory of redshift distortion of the galaxy power
spectrum in the discrete wavelet transform (DWT) representation. Because the
DWT power spectrum is dependent of both the scale and shape (configuration) of
the decomposition modes, it is sensitive to distortion of shape of the field.
On the other hand, the redshift distortion causes a shape distortion of
distributions in real space with respect to redshift space. Therefore, the
shape-dependent DWT power spectrum is useful to detect the effect of redshift
distortion. We first established the mapping between the DWT power spectra in
redshift and real space. The mapping depends on the redshift distortion effects
of (1) bulk velocity, (2) selection function and (3) pairwise peculiar
velocity. We then proposed $\beta$-estimators using the DWT off-diagonal power
spectra. These $\beta$-estimators are model-free even when the non-linear
redshift distortion effect is not negligible. Moreover, these estimators do not
rely on the assumption of whether the pairwise velocity dispersion being
scale-dependent. The tests with N-body simulation samples show that the
proposed $\beta$-estimators can yield reliable measurements of $\beta$ with
about 20% uncertainty for all popular dark matter models. We also develop an
algorithm for reconstruction of the power spectrum in real space from the
redshift distorted power spectrum. The numerical test also shows that the real
power spectrum can be well recovered from the redshift distorted power
spectrum.
