In the simplest applications, solutions can be most easily found in the frequency domain. When complications arise, it is better to use the time domain, to directly apply the convolution operator and the method of least squares. A first complicating factor in the frequency domain is a required boundary in the time domain, such as that between past and future, or requirements that a filter be nonzero in a stated time interval.
Weighting functions are appropriate whenever a signal or image amplitude varies from place to place.
Most of the literature on time-series analysis applies to the limited case of uniform weighting functions. This means that their statistical properties do not change in time. In real life, particularly in the analysis of echos, signals are never stationary in time and space. A stationarity assumption is a reasonable starting assumption, but we should know how to go beyond it so we can take advantage of the many opportunities that do arise. A time boundary such as between past and future.