In order to talk about an excess, it is obviously important to define compared to what the excess is measured. The body of literature on the faint galaxy excess problem is considerable, but different studies have used different references and this has led to confusing claims of various amounts of evolution over the past half Hubble time. Galaxy luminosity functions (LFs) are usually expressed following Schechter schech76:
where L is the galaxy luminosity and 
dL is the number of galaxies with luminosity between L and L+dL per Mpc
. Equation 
can be rewritten in absolute magnitude form as:
where M is the galaxy absolute magnitude and 
dM is the number of galaxies with magnitude between M and M + dM per Mpc. The Schechter function has three free parameters which must determined empirically: 
, 
and M
. The slope of the LF at the faint end is determined by 
. A luminosity function with 
= 
1.0 is said to be flat. M is called the characteristic Schechter luminosity, and is the number of galaxies per Mpc per magnitude at the characteristic luminosity.
It turns out that there is considerable uncertainty in the local values of 
and , and the root of this uncertainty is still not known. Part of the problem is that the parameters may be dependent upon galaxy morphological types, and the exact contribution of each galaxy type to the global LF is not well-known. The situation at higher redshifts is even worse as different galaxy types may evolve differently and selection effects may favor the detection of certain galaxy types over others.
The Stromlo-APM survey [\protect\astronciteLoveday et al.1992] studied a local sample of 1769 galaxies complete down to a magnitude limit of b
= 17.15. The sample was drawn at random from the APM Bright Galaxy Catalog. They found that the local luminosity function was well fitted over the magnitude range 22 < M
< 15 by a Schechter function with parameters M
= 19.50, 
= 0.97 and = 1.40
10
Mpc
. Hence, the local luminosity function appeared to be flat.
The Las Campanas Redshift Survey [\protect\astronciteLin et al.1996] covered 18678 galaxies with an average redshift of z = 0.1. The LCRS luminosity function could be fitted by a Schechter luminosity function with M = 20.29 + 5 log h (h 
H
/100), = 1.910 Mpc and 
= 0.7 over the absolute magnitude range 23.0 < M5 log h < 17.5. Moreover, the LCRS found that emission ([OII] equivalent width > 5 Å) and non-emission galaxies had different luminosity functions, with emission galaxies dominating the faint end and non-emission galaxies prevailing at the bright end.
Both the Loveday et al. survey and the LCRS derived the same normalization for the local LF. However, when this normalization is used in models trying to reproduce faint number counts of galaxies, these models fall short by a factor of 2 at relatively bright magnitudes (B 
1618). Since it is believed that early-type galaxies are old, dynamically relaxed systems, it is hard to understand why half of them would have disappeared over the past 5-6 Gyrs. Therefore, many faint galaxy studies have normalized their number count models to the observed counts at B 1618. Any excess observed at fainter magnitudes (B 2224) is measured over observed bright counts and not over the local luminosity functions.
The CfA Redshift Survey [\protect\astronciteMarzke et al.1994b] covered 9063 galaxies with Zwicky m
magnitude < 15.5 to calculate the galaxy luminosity function over the range 13 < M < 22. For galaxies with velocities cz > 2500 km s
, the luminosity function was well-represented by a Schechter function with parameters = 4.0 10 Mpc, M = 18.8 and 
= 1.0. The normalization was a factor of 2 higher than found by the Stromlo-APM and LCRS surveys. When all galaxies with cz > 500 km s were included, the number of galaxies in the range 16 < M < 13 exceeded the extrapolation of the Schechter function by a factor of 3 i.e. the LF rose steeply at magnitudes fainter than 16. This steep faint end excess was dominated by Magellanic spirals and irregulars, and their LF had M = 18.79, 
= 1.87 and = 0.610 Mpc [\protect\astronciteMarzke et al.1994a]. This abundance of intrinsically faint, blue nearby galaxies will obviously affect the magnitude of the excess seen in faint number counts and could go a long way in reconciling no-evolution standard models with faint number counts. It is not known at the present time why the steep faint end of the CfA local LF has not been detected by the Stromlo-APM and LCRS surveys.
Given the uncertainty in the local LF, it is reassuring to know that, as explained later, internal kinematics studies will not rely on the local LF to calculate the amount of luminosity evolution in intermediate redshift galaxies. However, the choice of local LF should be explicitly stated in any comparison between luminosity evolution measured from luminosity functions and internal kinematics.
