The resolvability and the capacity of AWGN channels are equal

The authors have introduced the new notion of resolvability of a channel, as the dual of the capacity, which is defined as the minimum complexity per input letter needed to generate an input process whose output distribution via the channel arbitrarily accurately approximates any prescribed achievable output distribution. The resolvability thus introduced has revealed a deep relationship between the minimum achievable rate for source coding, the channel capacity, the identification capacity and the problem of random number generation. However, the validity of the proof of the converse for the resolvability formula established by Han and Verdu (see IEEE Trans. on Inform. Theory, vol.39, no.3, p.379, 1993) hinged heavily on the assumption that the input alphabet of the channel is finite. Our main purpose in this paper is to show that we can relax this restriction and to show that the resolvability formula of Han and Verdu continues to hold also for a wide class of channels with continuous input alphabet, including as a special case, additive white Gaussian noise (AWGN) channels with power constraint.