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

TL;DR

This paper introduces superfunctions as a new class linked to subspace collections, establishing their relationship with rational functions of several variables and exploring operations and normalization without inner products.

Contribution

It develops the theory of superfunctions and subspace collections, showing their correspondence with rational functions and extending normalization and reduction operations beyond inner product spaces.

Findings

Every homogeneous degree 1 rational matrix function can be generated from a subspace collection.

Operations on subspace collections correspond to algebraic operations on rational functions.

Superfunctions may contain more physical information than functions, aiding in problem analysis.

Abstract

A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of subspace collections. Operations on subspace collections are found to correspond to various operations on rational functions, such as addition, multiplication and substitution. It is established that every rational matrix valued function which is homogeneous of degree 1 can be generated from an appropriate, but not necessarily unique, subspace collection: the mapping from subspace collections to rational functions is onto, but not one to one. For some applications superfunctions may be more important than functions, as they incorporate more information about the physical problem, yet can be manipulated in much the same way as functions. Previously subspace collections had been introduced when there was an inner product on the vector (or Hilbert) space, and appropriate subspaces were mutually orthogonal. In that setting certain normalization and reduction operations on subspace collections led to a continued fraction expansion of the associated function, which allowed one to bound the function in terms of a set of weight matrices and normalization matrices that are derived from series expansions. Here we also initiate the theory of normalization and reduction operations, appropriate when there is no inner product on the space.