  We present a theoretical analysis of the approximation properties of
convolutional architectures when applied to the modeling of temporal sequences.
Specifically, we prove an approximation rate estimate (Jackson-type result) and
an inverse approximation theorem (Bernstein-type result), which together
provide a comprehensive characterization of the types of sequential
relationships that can be efficiently captured by a temporal convolutional
architecture. The rate estimate improves upon a previous result via the
introduction of a refined complexity measure, whereas the inverse approximation
theorem is new.
