  This article presents an overview of a theory for performing temporal
smoothing on temporal signals in such a way that: (i) temporally smoothed
signals at coarser temporal scales are guaranteed to constitute simplifications
of corresponding temporally smoothed signals at any finer temporal scale
(including the original signal) and (ii) the temporal smoothing process is both
time-causal and time-recursive, in the sense that it does not require access to
future information and can be performed with no other temporal memory buffer of
the past than the resulting smoothed temporal scale-space representations
themselves.
  For specific subsets of parameter settings for the classes of linear and
shift-invariant temporal smoothing operators that obey this property, it is
shown how temporal scale covariance can be additionally obtained, guaranteeing
that if the temporal input signal is rescaled by a uniform scaling factor, then
also the resulting temporal scale-space representations of the rescaled
temporal signal will constitute mere rescalings of the temporal scale-space
representations of the original input signal, complemented by a shift along the
temporal scale dimension. The resulting time-causal limit kernel that obeys
this property constitutes a canonical temporal kernel for processing temporal
signal in real-time scenarios when the regular Gaussian kernel cannot be used
because of its non-causal access to information from the future and we cannot
additionally require the temporal smoothing process to comprise a complementary
memory of the past beyond the information contained in the temporal smoothing
process itself, which in this way also serves as a multi-scale temporal memory
of the past.
  This theory is generally applicable for both: (i) modelling continuous
temporal phenomena over multiple temporal scales and (ii) digital processing of
measured temporal signals in real time.
