  Designing and implementing systems as an interconnection of smaller
subsystems is a common practice for modularity and standardization of
components and design algorithms. Although not typically cast in this
framework, many of these approaches can be viewed within the mathematical
context of functional composition. This paper re-interprets and generalizes
within the functional composition framework one such approach known as filter
sharpening, i.e. interconnecting filter modules which have significant
approximation error in order to obtain improved filter characteristics. More
specifically, filter sharpening is approached by determining the composing
polynomial to minimize the infinity-norm of the approximation error, utilizing
the First Algorithm of Remez. This is applied both to sharpening for FIR,
even-symmetric filters and for the more general case of subfilters that have
complex-valued frequency responses including causal IIR filters and for
continuous-time filters. Within the framework of functional composition, this
paper also explores the use of functional decomposition to approximate a
desired system as a composition of simpler functions based on a two-norm on the
approximation error. Among the potential advantages of this decomposition is
the ability for modular implementation in which the inner component of the
functional decomposition represents the subfilters and the outer the
interconnection.
