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in three previous papers we have developed a method of measuring galaxy power spectrum with discrete wavelet transform ( dwt ) decomposition , which is an alternative of the fourier power spectrum detection ( fang & feng , 2000 ( paper i ) , yang et al 2001a ( paper ii ) , 2001b ( paper iii ) ) . the dwt power spectrum estimator is information - loseless , and optimized in the sense that the spatial resolution is adaptive automatically to the scale to be studied . a test with observed sample of the lcrs galaxies showed that the dwt estimator can give a robust measurement of the power spectrum .    in this paper , we continue our effort in this direction . the topic this time is to develop theory and algorithm of redshift distortion in the dwt representation .    in terms of power spectrum measurement , the central problem of redshift distortion is to find the mapping between the power spectra in redshift and physical spaces . in our paper ii , we have already studied this mapping . however , that mapping is directly obtained by a wavelet transform of the mapping of the power spectrum in the fourier representation ( peacock & dodds 1994 ) . although a fourier mode can be transformed into dwt modes and _ vice versa _ , we should be careful in doing the transform of a second order statistical quantity from the fourier representation into the dwt one . as we have showed in paper i , the covariance of density contrast in the dwt representation @xmath1 is not equivalent to the fourier counterpart @xmath2 , because the fourier mode is subjected to the central limit theorem , while mode localized in both scale and physical space may not be so . this is because the dwt mode is characterized by not only the scale , but also the configuration ( or shape ) of the mode . simply speaking , the variance @xmath1 contains information of both the scale and phase ( shape ) , while @xmath3 does not contain information of phases ( fang & thews 1998 ) . we have found in paper ii that , the fourier mapping of redshift distortion can be employed for diagonal dwt power spectrum , but not off - diagonal dwt power spectrum . this is due to the shape(phase)-dependence of dwt modes . a 3-d fourier mode with wave - vector @xmath4 can be transformed into mode @xmath5 by a coordinate rotation as long as @xmath6 . therefore , for an isotropic random field , the fourier modes with the same @xmath7 are statistically equivalent . however , the dwt modes do not share the same property . the length scale of a 3-d wavelet mode with scale index @xmath8 is @xmath9^{1/2}$ ] , where @xmath10 is the length scale of the sample . generally , one can not transform a mode @xmath11 to @xmath12 by a rotation , even if they have the same scale .    as a consequence , the redshift distortion in the dwt presentation will be shape - dependent . for instance , in the fourier representation , the linear redshift distortion factor @xmath13 ( kaiser 1987 ) , where @xmath0 is the so - called redshift distortion parameter , depends only on @xmath14 , i.e. the cosine of the angle between the wavevector @xmath15 and the line - of - sight . while in the dwt representation , the linear redshift distortion factor will depend on not only the scale , but also the shape of dwt modes . an important application of redshift distortion is to determine the redshift distortion parameter @xmath0 ( e.g. hamilton 1998 ) , which contains valuable information of the cosmological mass density parameter and the bias of galaxies . the shape - dependence of redshift distortion is very useful for the parameter determination . we will develop an algorithm of @xmath0 estimation with diagonal and off - diagonal dwt power spectra . moreover , the dwt representation provides an easy way to study the effect of selection function on the redshift distortion . in the first three papers , we assumed that the selection function @xmath16 is known . actually , the  known " is not necessary . according to the definition , selection function is an observed galaxy distribution if galaxy clustering is absent . therefore , for a consistent algorithm of power spectrum , the selection function should be and can be obtained by the galaxy distribution itself . we will show that in the dwt representation the selection function can be determined by the distribution of galaxies without other assumption . this algorithm is convenient to estimate the contribution of selection function to redshift distortion . the paper will be organized as follows . 2 is a summary of the algorithm of the dwt power spectrum . in  3 , we develop a theory of the redshift distortion with a multiresolution analysis . the emphases are the real - redshift mapping of diagonal and off - diagonal power spectrum . we proposed @xmath0-estimators , which are tested by n - body simulation samples of popular dark matter models (  4 and 5 ) . finally , the conclusions and discussions will be presented in  6 . the mathematical stuffs with the relevant dwt quantities are given in appendix . we summarize the algorithm of the dwt power spectrum . the details can be found in fang & feng ( 2000 ) and yang et al ( 2001 ) . if the position measurement is perfectly precise , the number density distribution of galaxies can be written as @xmath17\ ] ] where @xmath18 is the total number of galaxies , @xmath19 is position of @xmath20th galaxy , @xmath21 is its weight , and @xmath22 the 3-d dirac @xmath23 function . @xmath16 is selection function given by the mean number density of galaxies when galaxy clustering is absent , and @xmath24 is the density contrast fluctuation in the underlying matter distribution .    without loss of generality , we enclose the sample in a cubic box with side lengths @xmath25 . in the dwt representation , @xmath26 is decomposed into @xmath27 where , @xmath28 is given by @xmath29 the index @xmath8 stand for 3-d scales @xmath30 , and index @xmath31 for the position of cell @xmath32 , @xmath33 , @xmath34 .    in eq.(3 ) , @xmath35 is called scaling function coefficient ( sfc ) of @xmath26 . they are given by @xmath36 where we have used equation ( 1 ) in the last step . the scaling functions @xmath37 play the role of window functions for the cell with volume @xmath38 , and located at position @xmath39 . therefore , the sfc @xmath35 is the mean of field @xmath26 in the volume @xmath38 at position @xmath39 . thus , the term @xmath28 of eqs.(2 ) and ( 3 ) actually is the smoothed @xmath28 by window on scale @xmath40 . the 3-d scaling functions @xmath37 can be constructed by a direct product of 1-d scaling functions , i.e. @xmath41    in eq.(2 ) , @xmath42 are wavelets . an 1-d wavelet @xmath43 is the modes used to extract the fluctuations of a 1-d field @xmath44 on scale @xmath45 at position @xmath39 . the 3-d wavelets @xmath42 are given by mixed direct products of 1-d scaling functions @xmath46 and wavelets @xmath47 as follows , @xmath48 accordingly , @xmath49 with @xmath50 describe 2-d projected fluctuations on a slice , and @xmath51 1-d fluctuations along a line . @xmath52 describes the 3-d fluctuations inside the cell @xmath53 on a scale @xmath54 . hereafter , we consider the projection of density fluctuations onto the space spanned by @xmath52 only . for simplicity , we ignore the upper index @xmath55 in the notations of @xmath49 and @xmath56 without confusion . thus , the wavelet function coefficient ( wfc ) @xmath57 of @xmath26 is given by @xmath58 therefore , the wfc @xmath57 is the amplitude of the fluctuations of the field @xmath26 on scale @xmath38 at position @xmath39 .    the functions @xmath37 and @xmath59 form a complete and orthonormal basis in space of functions defined in 3-dimensional coordinate space . the decomposition of eq.(2 ) is information loseless . the dwt power spectrum of @xmath24 can be estimated by @xmath60 ^ 2}{n^2({\bf j , l})}\right \rangle   -\left \langle \frac{1}{n({\bf j , l})}\right \rangle,\ ] ] where @xmath61 is the average over ensemble . the first term on the r.h.s . is the normalized power from @xmath26 , and the second term corrects for the poisson noise . the normalization factor @xmath62 is the mean selection function in the mode @xmath63 , i.e. , the number density of galaxies in the mode @xmath63 when galaxy clustering is absent . as has been shown in paper i ( fang & feng 2000 ) , the factor @xmath64 can be absorbed into the weight factor by @xmath65 thus , equation(8 ) yields @xmath66 ^ 2 - \frac{1}{2^{j_1+j_2+j_3}}\sum_{l_1=0}^{2^{j_1}-1 } \sum_{l_2=0}^{2^{j_2}-1}\sum_{l_3=0}^{2^{j_3}-1 } \int\frac{\psi_{\bf j , l}^2({\bf x})}{\bar{n}({\bf x } ) } d{\bf x},\ ] ] where the wfc @xmath67 is defined as @xmath68      by definition eq.(1 ) , selection function @xmath16 is an observed galaxy distribution if galaxy clustering @xmath69 is absent . that is , equation ( 1 ) requires to decompose an observed distribution @xmath26 into two parts : the `` background '' @xmath16 , and the fluctuations @xmath24 upon the background . this decomposition is not easy if the selection function @xmath16 is position - dependent , i.e. it will mix with the fluctuations to be detected . with the dwt analysis , one is capable of performing this decomposition by a scale - by - scale analysis .    when we study the fluctuations on a scale @xmath40 , all @xmath70-dependencies of @xmath26 on scales larger than this scale play the role as a background . thus , in terms of the scale @xmath40 , the background is given by a smooth of @xmath26 on the scale @xmath40 , i.e. it does not contain information of fluctuations on scales equal to or less than @xmath40 . this background has already be found from equations ( 2 ) and ( 3 ) . because of the orthogonal relation @xmath71 the function @xmath28 does not contain any information of fluctuations of modes @xmath72 with @xmath73 . thus , to detect the fluctuation power on the scale @xmath40 , the function @xmath28 is recognized as a background field . one can identify the selection function as @xmath28 , i.e. @xmath74 where the superscript @xmath40 means that this `` selection function '' is only for the scale @xmath40 .    in the plane parallel approximation , selection function depends on @xmath75 only , i.e. the coordinate in redshift direction . from equation ( 13 ) , the selection function is given by @xmath76 from equation ( 6 ) , the sfc @xmath77 is actually given by an average of @xmath26 over the plane @xmath78 , and decomposition along @xmath75 direction . using equation ( 14 ) , the wfc @xmath67 is now calculated by @xmath79 which presents a simple algorithm for deriving the selection function from observed galaxy samples . to test this algorithm , we produce mock galaxy samples using n - body simulation , of which the details will be given in  4 . for simplicity , the selection effect is applied along one axis ( e.g. , @xmath75 direction ) of 3-dimensional cartesian coordinates under the plane parallel approximation . for the simulation sample in the cubic box with a side length of 256 h@xmath80 mpc , we replicate the sample along @xmath75 direction , and choose the mock galaxies located between 100 - 356 h@xmath80 mpc . a selection function is given by @xmath81 where @xmath10 is the size of the sample . to be comparable with , for example , the lcrs selection function , the parameters are adopted to be @xmath82 h@xmath80 mpc , @xmath83 and @xmath84 . 1 displays the dwt diagonal power spectrum @xmath85 of particle distributions for three typical models with the selection function eq.(14 ) , in which we take @xmath86 . it shows that the power spectrum estimator eqs.(10 ) and ( 15 ) can perfectly recover the dwt power spectrum regardless the selection functions ( 16 ) . this result will keep unchanged if @xmath87 . it means that @xmath88 gives a proper estimation of selection function if @xmath89 is high enough . practically , one can find a properly recovered power spectrum by checking whether it is insensitive to @xmath89 . it should be pointed out that with the developed algorithm the power spectrum eq.(8 ) is normalized scale - by - scale . the fluctuation amplitudes @xmath90 ( wfcs ) on scale @xmath40 is normalized by @xmath62 or @xmath91 [ eq.(15 ) ] , which contains fluctuations on all scales larger than @xmath40 . therefore , the normalization factor generally is scale - dependent . this is different from conventional normalization , which is scale - independent . if the field is gaussian , there is no correlation between @xmath90 and @xmath62 or @xmath91 . eq.(8 ) yields @xmath92 ^ 2\right \rangle -    \left \langle \frac{1}{n({\bf j , l})}\right \rangle.\ ] ] it has been shown with the so - called  partition of unity " of wavelets that @xmath93 is approximately independent of @xmath40 . in this case , the power spectrum eq.(8 ) is the same as that given by conventional normalization ( jamkhedkar , bi & fang 2001 ) . however , for non - gaussian field , especially , if the perturbations on different scales are correlated , the small scale fluctuations given by the wfc @xmath90 generally are correlated with large scale fluctuation contained in @xmath62 or @xmath91 . in this case , the power spectrum eq.(8 ) is very different from usual power spectrum which does nt not sensitive to the correlation between perturbations on different scales . the power spectrum defined by eq.(8 ) will benefit to calculate the effect of selection function upon the redshift distortion (  3.3 ) . the redshift distortion is due to peculiar motion of galaxies . for a given mass field @xmath24 , the galaxy velocity @xmath94 is a random field with mean @xmath95 where @xmath96 denotes ensemble average for velocity fields . @xmath97 is the bulk velocity at @xmath98 .    in linear regime , the bulk velocity is related to the density contrast by @xmath99 where @xmath100 is the redshift distortion parameter at present , i.e. redshift @xmath101 . the _ rms _ deviation of velocity @xmath102 from the bulk velocity @xmath97 is @xmath103 ^ 2\rangle_v    = \sigma^2_{pv}({\bf x}).\ ] ] in the dwt representation , one can calculate @xmath104 by the variance of the wfcs of velocity field , i.e. @xmath105 , which is actually the dwt pairwise peculiar velocity ( paper iii ) .    in the paper iii , we analyzed the pvd of the velocity fields given by the n - body simulation for the cdm family of models . on large scales , the velocity field is basically gaussian . it is completely described by its mean @xmath97 and variance @xmath106 . on small scales , the velocity field is significantly non - gaussian . the one point function of @xmath107 is lognormal . the two - point pairwise velocity correlation functions are non - zero on scales @xmath108 1 h@xmath80 mpc . moreover , on these scales , the pairwise velocities are also correlated significantly with density fluctuations . the position of galaxy @xmath20 in redshift space is given by @xmath109 , where @xmath110 is the radial component of @xmath102 . the number density distribution in redshift space is then @xmath111=    \bar{n}^s({\bf s})[1+\delta^s({\bf s})],\ ] ] where @xmath112 is the selection function in redshift space . similar to equation ( 8) , the power spectrum in redshift space is @xmath113 ^ 2 - \frac{1}{2^{j_1+j_2+j_3}}\sum_{l_1=0}^{2^{j_1}-1 } \sum_{l_2=0}^{2^{j_2}-1}\sum_{l_3=0}^{2^{j_3}-1 } \int\frac{\psi_{\bf j , l}^2({\bf x})}{\bar{n}^s({\bf x } ) }   d{\bf x},\ ] ] where @xmath114 the selection function @xmath112 can be determined from the observed distribution @xmath115 by equation ( 13 ) or ( 14 ) . it can then be absorbed into the weight @xmath21 as equation ( 11 ) or ( 15 ) . the effect of the difference between @xmath115 and @xmath116 will be studied in next section . in this section , we will ignored this effect . thus , using an auxiliary vector @xmath117 , equations ( 21 ) and ( 23 ) yield @xmath118 where @xmath119 is gradient operator on @xmath120 .    subjecting equation ( 24 ) to an average over ensemble of velocity , if the velocity field is _ gaussian _ , we have @xmath121 for simplicity , we will use @xmath122 for @xmath123 below without causing confusion . if we consider only linear effect of the bulk velocity , equation ( 24 ) gives @xmath124 e^{-(1/2)\sigma^2_{pv}({\bf x}_i)({\bf j}\cdot\hat{\bf r})^2 } \psi_{\bf j , l}({\bf s } )   \right |_{{\bf j}=0 } \\ \nonumber    & = &   \sum_{i=1}^{n_g}w_i   \int d{\bf s }   \psi_{\bf j , l}({\bf s } )     e^{-(1/2)\sigma^2_{pv}({\bf s})(\hat{\bf r}\cdot \nabla ) ^2 }     \delta_d ( { \bf s } -{\bf x}_i )   \\ \nonumber      & - &   \frac{1}{h_0}\sum_{i=1}^{n_g}w_i   \int d{\bf s } \left . \hat{\bf r }     \cdot [ \nabla_s v_r({\bf s}+i\nabla_j )      \delta_d \left ( { \bf s } -{\bf x}_i+ i \nabla_j \right ) ]     e^{-(1/2)\sigma^2_{pv}({\bf x}_i)({\bf j}\cdot\hat{\bf r})^2 }     \psi_{\bf j , l}({\bf s } )   \right |_{{\bf j}=0 } \\ \nonumber\end{aligned}\ ] ] neglecting the terms of the order of @xmath125 , and using the linear relation eq.(19 ) , equation ( 26 ) gives @xmath126 because all operators in the integrand of equation ( 27 ) are nearly diagonal in the dwt representation ( farge et al 1996 ) , equation ( 27 ) can be rewritten as @xmath127 where @xmath128 and @xmath129 the method of calculating @xmath130 and @xmath131 is presented in appendix a.    substituting equation ( 28 ) into equation ( 22 ) , we have the redshift distorted power spectrum as @xmath132 where @xmath133 ^ 2 $ ] . above equation formulates the redshift distortion effect in dwt expression . this is the basic formula for our redshift distortion analysis . usually , the factor @xmath134 is called linear redshift distortion , and @xmath135 called non - linear redshift distortion due to the pairwise velocity dispersion . however in our derivation , the two parts are not treated separately . this derivation can be generalized to any velocity fields , which are not simply described by equations ( 18 ) and ( 20 ) . for instance , if the pairwise velocities are correlated , i.e. @xmath103[v_i({\bf x'})-v_i({\bf x'})]\rangle_v = \sigma^2_{pv}({\bf x - x'}),\ ] ] the pairwise velocity dispersion factor becomes @xmath136 }     \psi_{\bf j , l}({\bf x } ) \psi_{\bf j , l}({\bf x'}),\end{aligned}\ ] ] where @xmath137 is gradient operator on @xmath138 . the theory of redshift distortion presented in last section did not consider the effects of selection functions , @xmath139 and @xmath140 . since selection function relies on the radial distance , it could be also a source of the anisotropy of power spectrum with respect to redshift direction . it should be taken into account when analyze observed samples .    in the linear approximation of @xmath110 , equation ( 21 ) gives @xmath141.\ ] ] in the second steps , @xmath110 is replaced by @xmath142 , as only the bulk velocity is considered . using equations ( 1 ) and ( 21 ) , equation ( 34 ) yields @xmath143 \right \},\ ] ] where the second order term @xmath144 is ignored . the term -1 on the r.h.s . of equation ( 35 ) does not contribute to power spectrum , because of @xmath145 . in the linear approximation , @xmath146 and therefore , the factor @xmath147 in the third term of the r.h.s . of equation ( 34 ) can be approximated as 1 .    the second term in the r.h.s . of equation ( 35 ) contains a linear term of @xmath142 , i.e. the same order as the third term . however , with the selection function equation ( 13 ) , we have @xmath148 for plane parallel approximation , @xmath149 , the above equation is always hold if we study fluctuation powers on scales @xmath150 or @xmath151 . actually , in this case , @xmath152 or @xmath153 depend only on @xmath75 , and thus their projections onto bases @xmath154 or @xmath155 ( @xmath156 ) are always null . similarly the selection function in redshift space is also only dependent on @xmath75 . accordingly , in the plane parallel approximation , we have @xmath157 which implies that the second term in the r.h.s . of equation ( 34 ) also has not contribution to the projection on bases @xmath158 or @xmath159 . in fact , the condition of plane parallel approximation is not necessary . for radial redshift , the selection function is still a 1-d function . its projection onto the bases @xmath160 on the celestial spherical surface is still zero . thus , the linear redshift distortion mapping is given by @xmath161.\ ] ] hereafter we will use @xmath98 for the variable @xmath162 . it will not cause confusion as the superscript @xmath163 stands for redshift space .    in the plane parallel approximation , equation ( 38 ) becomes @xmath164 using equation ( 19 ) , @xmath165 can be represented by @xmath69 , we have then @xmath166 \delta({\bf x}).\ ] ] the differential operator of the second term in the bracket of equation ( 40 ) is nearly diagonal in the dwt representation ( farge et al 1996 . ) thus , we have @xmath167 where @xmath168 means the value of @xmath169 in the mode @xmath170 . the coefficient @xmath171 are defined by @xmath172 the calculations of @xmath171 are given in appendix a.    since @xmath173 ( appendix a ) , the first and second terms on the r.h.s . of equation ( 40 ) are not correlated , we have @xmath174 ^ 2      \langle |\tilde{\epsilon}_{\bf j , l}|^2   \rangle +    \left [ \left . \beta\frac{d\ln \bar{n}(x_3)}{dx_3}\right|_{\bf j , l}\right ] ^2      \sum_{l_3-l'_3 } q^2_{{\bf j } , l_3-l'_3 }        \langle |\tilde{\epsilon}_{{\bf j},l_1,l_2,l'_3}|^2   \rangle\ ] ] for uniform fields , @xmath175 and @xmath176 are @xmath39-independent . hence , equation ( 43 ) gives the relation between the dwt power spectra in redshift @xmath177 and real spaces @xmath178 as @xmath179 ^ 2 +   \left [ \left . \beta\frac{d\ln \bar{n}(x_3)}{dx_3}\right|_{\bf j , l}\right ] ^2   \sum_{l_3-l'_3 } q^2_{{\bf j } , l_3-l'_3 } \right \}p_{\bf j}.\ ] ] which quantifies both redshift distortion and selection effect on the dwt power spectrum .    using inequality equation ( a15 ) , we can show that if @xmath180 we have @xmath181 ^ 2   \sum_{l_3-l'_3 } q^2_{{\bf j } , l_3-l'_3 } < \beta^2 s^2_{\bf j}.\ ] ] which states that the selection function term in equation ( 44 ) is even less than the second order terms @xmath182 if the selection function varies with @xmath75 slowly . in this case , the selection function does not significantly disturb the linear redshift distortion described by equation ( 31 ) when we perform a dwt power spectrum analysis using equations ( 10 ) and ( 15 ) .    the poisson noise term in equation ( 10 ) is not affected by the redshift distortion , i.e. it is the same as the poisson noise term in equation ( 22 ) . because both poisson noise terms of equations ( 10 ) and ( 22 ) linearly depend on @xmath183 or @xmath184 , for an ensemble average , we have @xmath185 . to demonstrate the redshift distortion in the dwt representation , we use the n - body simulation samples like that in paper ii . the model parameters used are listed in table . 1 . we use modified ap@xmath186 m code ( couchman , 1991 ) to evolve @xmath187 cold dark matter particles in a periodic cube of side length @xmath10 . the linear power spectrum is using the fitting formula given in bardeen et al . ( 1986 ) . table 1    [ cols="^,^,^,^,^,^,<,^",options="header " , ]     in our simulation , we use the so - called  glass " configuration to generate the unperturbed uniform distribution of particles , and the zeldovich approximation to set up the initial perturbation . the triangular - shaped cloud ( tsc ) method is used for the mass assignment on the grid and the calculation of the force on a given particle from interpolation of the grid values . we take 600 total integration steps from @xmath188 for the scdm model , and @xmath189 for @xmath190cdm and @xmath191cdm down to @xmath192 . the force softening parameter @xmath193 in the comoving system decreases with time as @xmath194 . its initial value is taken to be 15% , and the minimum value to be 5% of the grid size , respectively . we try to estimate the redshift distortion parameter @xmath0 using equation ( 30 ) . obviously , the precision of the @xmath0 determination is dependent on the linear relation of equation ( 18 ) , which is valid only on the scales where the bulk velocity can be described by linear or quasi - linear density perturbations . it has been already realized that the non - linearity of the relation between @xmath24 and @xmath97 will be significant on small scales ( e.g. kudlicki et al . therefore , it is necessary to have an estimation of uncertainties due to the non - linear @xmath195 relation . the non - linear effect can be estimated by a  blueshift " ( change the sign of velocity from @xmath102 to @xmath196 ) distorted power spectrum . from the derivation of equation ( 31 ) in last section , it is easy to show that , the blueshifted power spectrum is @xmath197 which has the same non - linear redshift distortion effect @xmath135 as the redshift power spectrum . the difference between the redshift [ equation ( 31 ) ] and blueshift [ equation ( 47 ) ] distorted power spectra is a sign of the linear term @xmath0 . combining equations ( 31 ) and ( 47 ) , one can determine @xmath0 as @xmath198    we test equation ( 48 ) by simulation samples in the @xmath190cdm model . first , we calculate the diagonal power spectra of particles , @xmath199 and @xmath200 , in the plane parallel approximation . we then find @xmath0 by equation ( 48 ) . the result is presented in fig . the density parameter used for the simulation is @xmath201 , or @xmath202 . 2 shows that the values of @xmath0 given by equation ( 48 ) is generally overestimated . but the overestimations are no more than 10% on scales @xmath203 h mpc@xmath80 , and about 20% for @xmath204 h mpc@xmath80 . it should be pointed out that the goodness of the resulted @xmath0 on small scales @xmath204 h mpc@xmath80 does nt mean that the linear relation eq.(19 ) can be used to describe the redshift distortion on these scales . the non - linear effect on these scales can not be ignored , and the redshift distortion is dominated by the term @xmath205 . the goodness for @xmath204 h mpc@xmath80 shown in fig . 2 is due to that the non - linear effect is significantly repressed by the test eq.(48 ) based on an assumed blueshifts .      at the first glance , it seems impossible to estimate @xmath0 using the observed redshift distorted power spectrum @xmath177 only , as all the quantities such as @xmath0 , @xmath178 , and @xmath135 ( or @xmath206 ) included on the r.h.s . of equation ( 30 ) are unknown . usually , equation ( 30 ) is used to fit the observed power spectrum with model - predicted @xmath178 , and then determine the parameter @xmath0 from the most - likely - fitting . but we try to search for @xmath0-estimators which depend on model as less as possible . to achieve this , we take the advantage of the wavelet analysis : the dwt modes of @xmath207 are not rotational invariant , but cyclic permutational invariant . as a consequence , all the dwt quantities in the redshift distortion equation ( 30 ) , @xmath178 , @xmath177 , @xmath130 , and @xmath135 , are dependent on the three indexes @xmath208 , rather than the length scale of mode @xmath40 only . the quantities @xmath178 , @xmath177 , @xmath130 and @xmath135 satisfy the following symmetry . \1 . if cosmic density and velocity fields are statistically isotropic , the dwt power spectrum in real space is invariant with respect to the cyclic permutations of index @xmath209 , i.e. @xmath210    \2 . in the plane - parallel approximation , e.g. , coordinate @xmath75 is in the redshift direction , we have @xmath211 @xmath212 @xmath213    \3 . if @xmath206 is constant , i.e. scale - independent , following equation ( a10 ) , we have @xmath214 i.e. @xmath135 is independent of @xmath215 and @xmath216 . more generally , equation ( 53 ) also hold even when the radial correlation of pairwise peculiar velocity is considered [ equation ( 33 ) ] . equations ( 49 ) - ( 53 ) provide the base of designing the @xmath0-estimators with the dwt power spectrum .      assuming that the pairwise velocity dispersion is scale - independent , equations ( 31 ) and ( 49 ) give @xmath217 ^ 2.\ ] ] for a given pair @xmath218 , @xmath219 is a constant , and thus the r.h.s . of equation ( 54 ) depends only on two parameters @xmath0 and @xmath219 . accordingly , these parameters can be found by the best fitting of the r.h.s . of equation ( 54 ) with observed ratios @xmath220 , @xmath221 . a numerical example of this fitting is demonstrated in fig . 3 , in which we take @xmath222 , and @xmath223 . we calculate @xmath220 for the @xmath190cdm simulation samples , and the best fitting yields the values of @xmath224 and @xmath225 . the precision of the estimator eq.(53 ) probably is not better than 20% . this shows that the assumption of a constant @xmath206 is not too bad , but its effect on the @xmath0-determination can not be neglected . if @xmath206 is scale - dependent , equation ( 54 ) is no longer correct in general . in this case @xmath226 depends on the three index @xmath208 , rather than @xmath89 only . so the relation eq.(30 ) is obviously not enough to extract @xmath0 from measuring redshift distorted power spectrum @xmath177 only . to search for an appropriate algorithm for @xmath0 estimation , we shall first consider the property of @xmath135 .    at first , the pairwise velocity dispersion factor @xmath135 is determined by an isotropic function @xmath106 , which is scale - dependent . let us consider the modes of @xmath227 , where @xmath228 is defined by @xmath229 . in this case , the scales of these modes are dominant by @xmath89 , and therefore , @xmath106 is also dominant by @xmath89 . that is , for a given @xmath89 , @xmath135 will keep constant if @xmath230 .    to test this expectation , we calculate @xmath135 in the @xmath190cdm model . the result is shown in fig . 4 . one can see from fig . 4 that @xmath135 is generally dependent on indexes @xmath228 as well as @xmath89 . however , for a given @xmath89 , @xmath135 almost keeps constant in the range of @xmath231 . since @xmath232 is the two largest scales of the samples , @xmath233 and @xmath234 keep constant very well for all @xmath235 .    using this result , we can design a @xmath0-estimator as follows @xmath236 or @xmath237   \frac{1}{(s_{j23}-s_{j32}+s_{j'32}-s_{j'23})}\ ] ] obviously , the first factor on the r.h.s . of equation ( 56 ) is to measure the difference between spectra with the same scales , but different shapes of the modes . the estimator eq.(56 ) is model - free , as equation ( 56 ) is based only on property that two dwt modes of @xmath238 and @xmath239 have the same scale , but different shapes if @xmath231 and @xmath240 .    to apply the estimator ( 56 ) , we take @xmath241 and @xmath242 , because modes with @xmath241 and @xmath242 have largest difference in the shape , but very small difference in the scale . this estimator yields the values of @xmath243 for @xmath190cdm sample with simulation box @xmath244 480 h@xmath80 mpc ; @xmath245 for scdm sample with @xmath244 256 h@xmath80 mpc ; and @xmath246 for @xmath191cdm sample with @xmath244 256 h@xmath80 mpc . considering the uncertainty of non - linearity is about 20% (  5.1 ) , the estimator eq . ( 56 ) works very well . after the @xmath0 estimation , we can also estimate @xmath206 on scale @xmath89 using off - diagonal power spectrum in redshift space . from 5.4 , we know that @xmath135 is almost constant in the range of @xmath231 . so we have @xmath247 and @xmath248 . we can calculate the ratio of pairwise velocity dispersion factor @xmath249 by @xmath250 on the largest scale @xmath251 , the nonlinear redshift distortion due to the pairwise velocity dispersion is negligible , we have @xmath252 . thus , we have all the diagonal members of the pairwise velocity dispersion factor @xmath253 . the parameter @xmath206 on the scale @xmath254 can then be found by @xmath255 ^ 2 $ ] with equation ( a10 ) , i.e. @xmath256.\ ] ]    for the @xmath190cdm simulation sample , @xmath253 is shown in fig.5 . the @xmath206 on scales @xmath257 is shown in fig . although the values of @xmath206 shown in fig . 6 are correct in average , but it does not match with the direct measurement of @xmath206 given in paper iii . especially , on small scales @xmath258 6 and 7 , the values of @xmath206 are significantly lower than the direct measurement . this is not unexpected . the factor @xmath131 equation ( 58 ) is obtained under the assumption of gaussian velocity field (  3.1 ) . however , we have shown in paper iii that the velocity field is actually intermittent on small scales . the pdf of @xmath259 is not gaussian , but lognormal .    nevertheless , the factor @xmath135 is still good for the redshift - real mapping and @xmath0-estimation , because in these calculations , we used only the symmetric properties and scale - dependence of @xmath135 , but not the details of @xmath206 . this point can also be seen from reconstruction of power spectrum in real space from that in redshift space . using the @xmath0 estimated by equation ( 56 ) , and @xmath253 by equation ( 57 ) , one can reconstruct the diagonal dwt power spectrum in real space @xmath85 from @xmath199 through equation ( 31 ) . fig.(7 ) compares the recovery of @xmath85 with the original diagonal dwt power spectrum in the @xmath190cdm model . the result shows that the algorithm of reconstruction is reliable . we established the mapping between the dwt power spectra in real and redshift spaces . from equations ( 31 ) and ( 44 ) , the mapping in the plane parallel approximation is @xmath179 ^ 2 +   \left [ \left . \beta\frac{d\ln \bar{n}(x_3)}{dx_3}\right|_{\bf j , l}\right   ] ^2   \sum_{l_3-l'_3 } q^2_{{\bf j } , l_3-l'_3 } \right \}s^{pv}_{\bf j }      p_{\bf j}.\ ] ] which includes the effects of ( 1 ) bulk velocity ( the term of @xmath130 ) , and ( 2 ) selection function ( the term of @xmath260 ) as well as ( 3 ) pairwise peculiar velocity ( the term of @xmath135 ) .    in the fourier representation , the redshift distortion mapping is axially symmetric with respect to the redshift direction . this is because the fourier mode is rotational invariant , and the redshift distortion produces an anisotropy in the spatial distribution of galaxies between the line of sight and directions other than it . an isotropic statistics plus an axially symmetric violation will result in an axial symmetry . on the other hand , all redshift distortion factors in equation ( 59 ) are no longer axial symmetric , which is due to the modes @xmath40 being not invariant under rotational transformation , but for cyclical permutation . therefore , the symmetry of the dwt redshift distortion results from a cyclical permutation plus an axially symmetric violation .    by virtue of this feature , we develop @xmath0 estimators , which are mainly based on the shape - dependence of the redshift distorted dwt power spectrum , but not the scale - dependence . the estimators eqs.(54 ) and ( 56 ) look similar to the quadrupole - to - monopole ratio method based on the fourier power spectrum in redshift space , which gives a model - free estimation of @xmath0 ( e.g. cole et al , 1994 , fisher et al , 1994 , hatton & cole , 1998 , szalay et al , 1998 ) . however , the estimators eqs.(54 ) and ( 56 ) is different from the quadrupole - to - monopole ratio method . the latter considered only the linear redshift distortion effect , but not the non - linear redshift distortion effect . if considering the non - linear redshift distortion effect , the quadrupole - to - monopole ratio method needs a fitting of observed redshift distorted power spectrum with model - predicted power spectrum . on the other hand , the estimators eqs.(54 ) and ( 56 ) considered both linear ( @xmath261 ) and non - linear ( @xmath135 ) redshift distortion effects . these @xmath0-estimators are model - free even when the non - linear redshift distortion effect is not negligible . we test the @xmath0-estimators using n - body simulation samples . the result shows that regardless the pairwise velocity dispersion is scale - dependent or not , the @xmath0-estimators can yield the correct number of @xmath0 with about 20% uncertainty . we also develop an algorithm for reconstruction of the power spectrum in real space from the redshift distorted power spectrum . the numerical tests also show that the real power spectrum can be reasonably recovered from the redshift distorted power spectrum . llf and yqc acknowledges support from the national science foundation of china ( nsfc ) and national key basic research science foundation . let us consider plane - parallel approximation , i.e. coordinate @xmath75 is in the redshift ( @xmath263)-direction . the linear redshift distortion @xmath130 equation ( 28 ) gives @xmath264 because 1-d wavelet @xmath43 is given by dilating and translating the basic wavelet @xmath265 as @xmath266            if @xmath281 is independent of @xmath98 , @xmath282 [ equation ( 28 ) ] can be calculated by @xmath283,\ ] ] where vector @xmath284 . in plane - parallel approximation , we have @xmath285.\ ] ] the summation of equations . ( a9 ) and ( a10 ) also runs only over few number of @xmath275 around @xmath276 .                \\ \nonumber    & & \simeq \frac{l_3}{2\pi n_p2^{j_3 } } \frac{1}{2^{j_1+j_2+j_3 } }      \\ \nonumber    & &       \sum_{n_1,n_2,n_3 = \infty}^{\infty }    \frac{(n_3/l_3)^2\sin[2\pi n_3(l_3-l_3')/2^{j_3 } ] }     { [ ( n_1/l_1)^2 + ( n_2/l_2)^2 + ( n_3/l_3)^2 ] }
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 @xmath0-estimators using the dwt off - diagonal power spectra . 
 these @xmath0-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 @xmath0-estimators can yield reliable measurements of @xmath0 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 .