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tokens: velocities are conveniently expressed in terms of spherical Bessel functions, j l , and spherical Harmonics, Y lm . To do that, we imagine a very large spherical volume or radius R max which entirely encompasses the data and is also sufficiently large that the effects boundary conditions are small. Expressing the density contrast as where the wavenumbers k n are fixed by the boundary conditions at R max (Fisher et al. 1995 ). We advocate the boundary conditions which yield gravity potential decaying as r − l for r > R max . Since j l Y lm are eigenfunctions of the Laplacian, these boundary conditions are equivalent to finding k n which satisfy . The numbers C ln are fixed entirely by the boundary conditions and are given in table A1 of Fisher et al. ( 1995 ). The density coefficients are Using the linear velocity–density relation one obtains (Regos Szalay 1989 ; Scharf Lahav 1993 ; Fisher et al. 1995 ), where . For velocities measured with respect to the Local Group frame the l = 1 term should be modified by subtracting 1/3 from . Using this representation for u ( r ) yields where P l is Legendre polynomial of order l and we have used . From , the cross-correlation can easily be computed . Using the expression ( 7 ) which is appropriate for definition ( 2 ) we get 2 7 DISCUSSION An aim of this contribution is to show that different approaches to the determination of are tightly related. All methods yield something which mimics the behaviour of the theoretical bulk flow. The non-trivial challenge is to contrast any estimate with predictions of cosmological model. Systematic errors related to observables of galaxy properties, contaminate various methods in different ways. More distant galaxies are more prone to these systematics, therefore, consistency checks must be performed by applying the methods subsamples of the data. Furthermore, spatial coverage of the data depends on galaxy-type and unavoidable cuts on the observables. Thus, methods should be tested on mock galaxy catalogues that mimic the observations in as much as possible, including the observed large-scale structure. The constrained simulations (Sorce et al. 2013 ) designed to match the low-redshift large-scale structure, can potentially be highly beneficial if combined with galaxy formation models. We have also emphasized the differences between the methods. Hoffman et al. ( 2015 ) provide an estimate for the bulk flow as defined in equation ( 1 ), while Feldman et al. ( 2010 ) aim at given by equation ( 2 ). These two definitions of do not coincide (Nusser 2014 ) and should not produce identical results. None the less, both methods can easily be adapted to any of the two definitions. In Section 3 , we derive a new estimator which naturally arises from the likelihood . This is a generalization of the standard ML of Kaiser ( 1988 ), but incorporates velocity modes beyond the (constant) bulk flow of the entire data catalogue. Essentially, this estimator is equivalent to