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A natural connection between rational functions of several real or complexvariables, and subspace collections is explored. A new class of function,superfunctions, are introduced which are the counterpart to functions at thelevel of subspace collections. Operations on subspace collections are found tocorrespond to various operations on rational functions, such as addition,multiplication and substitution. It is established that every rational matrixvalued function which is homogeneous of degree 1 can be generated from anappropriate, but not necessarily unique, subspace collection: the mapping fromsubspace collections to rational functions is onto, but not one to one. Forsome applications superfunctions may be more important than functions, as theyincorporate more information about the physical problem, yet can be manipulatedin much the same way as functions. Previously subspace collections had beenintroduced when there was an inner product on the vector (or Hilbert) space,and appropriate subspaces were mutually orthogonal. In that setting certainnormalization and reduction operations on subspace collections led to acontinued fraction expansion of the associated function, which allowed one tobound the function in terms of a set of weight matrices and normalizationmatrices that are derived from series expansions. Here we also initiate thetheory of normalization and reduction operations, appropriate when there is noinner product on the space.

Superfunctions and the algebra of subspace collections and their association with rational functions of several complex variables