Mappings of window functions

In this paper we examine the mathematical relationships between window functions that share the same finite support. First, we find that the set of windows is closed under mappings that transform one into another or operations that operate on two or more windows that produce another window in the set. These operations include linear combinations, geometric combinations, raising to a power, and others, which all result in new window functions. The linear combination mapping operates on a number of windows to produce a new window in their convex hull and results in a search space using the Barycentric coordinates of any point to find optimal windows to satisfy a cost function. Any sequence of transformations or operations will result in fixed points. These fixed points, which are invariant under the operations, reflect the limits and importance of the operations. Operating on given windows with a sequence of transformations and operations, possibly iteratively, is shown to result in optimal windows.