  A continuous-time white Gaussian channel can be formulated using a white
Gaussian noise, and a conventional way for examining such a channel is the
sampling approach based on the classical Shannon-Nyquist sampling theorem,
where the original continuous-time channel is converted to an equivalent
discrete-time channel, to which a great variety of established tools and
methodology can be applied. However, one of the key issues of this scheme is
that continuous-time feedback cannot be incorporated into the channel model. It
turns out that this issue can be circumvented by considering the Brownian
motion formulation of a continuous-time white Gaussian channel. Nevertheless,
as opposed to the white Gaussian noise formulation, a link that establishes the
information-theoretic connection between a continuous-time white Gaussian
channel under the Brownian motion formulation and its discrete-time
counterparts has long been missing. This paper is to fill this gap by
establishing information-theoretic extensions of the Shannon-Nyquist theorem,
which naturally yield causality-preserving connections between continuous-time
Gaussian feedback channels and their associated discrete-time versions in the
forms of sampling and approximation theorems. As an example of the possible
applications of the extensions, we use the above-mentioned connections to
analyze the capacity of a continuous-time white Gaussian feedback channel.
