  We study the maximum achievable differential entropy at the output of a
system assigning to each input X the sum X+N, with N a given noise with
probability law absolutely continuous with respect to the Lebesgue measure and
where the input and the noise are allowed to be dependent. We consider fairly
general average cost constraints in the input, as well as amplitude
constraints. It is shown that the corresponding search for the optimum may be
performed over joint distributions for the input and the noise concentrated in
lower dimensional geometrical objects represented by graphs of sufficiently
regular functions in the associated noise-input plane. The results are then
applied to correspondingly characterize the independent input and noise case,
so providing bounds for channel capacity. Analysis of achievable bounds and
associated capacity-achieving input distributions is also provided. General
conditions for the existence of a maximum achievable entropy at the output are
provided for such a system, as well as for the independent input and noise
case, so establishing conditions for capacity to be achievable. The laws of
such capacity-achieving input distributions are also uniquely specified. A
general approximation to capacity scheme is also provided, showing that in the
not achievable case the corresponding noise can be written as the limit of a
sequence of noise random variables with associated channels being capacity
achievable, and whose capacities converge to the capacity of the original
channel. The relationship between our results for this input and noise
dependent setting and the notion of feedback capacity are also explored.
