  Lowpass envelope approximation of smooth continuous-variable signals are
introduced in this work. Envelope approximations are necessary when a given
signal has to be approximated always to a larger value (such as in TV white
space protection regions). In this work, a near-optimal approximate algorithm
for finding a signal's envelope, while minimizing a mean-squared cost function,
is detailed. The sparse (lowpass) signal approximation is obtained in the
linear Fourier series basis. This approximate algorithm works by discretizing
the envelope property from an infinite number of points to a large (but finite)
number of points. It is shown that this approximate algorithm is near-optimal
and can be solved by using efficient convex optimization programs available in
the literature. Simulation results are provided towards the end to gain more
insights into the analytical results presented.
