Frames for vector spaces and affine spaces

Shayne Waldron


A finite frame for a finite dimensional Hilbert space is simply a spanning sequence. We show that the linear functionals given by the dual frame vectors do not depend on the inner product (though they can be conveniently computed using it). Thus, it is possible to extend the frame expansion (and other elements of frame theory) to any spanning sequence for a finite dimensional vector space (over any subfield of the complex numbers which is closed under conjugation). The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact the associated Gramian matrix is an orthogonal projection. They depend continuously on the frame vectors, and transform naturally under linear maps. By way of contrast, the generalisations of the frame expansion to Banach spaces in the literature typically involve an analogue of the frame bounds and frame operator -- something which can completely be dispensed with in the finite dimensional case (where convergence is not an issue).

The same naive approach shows that for a finite dimensional affine space there are natural generalised barycentric coordinates for any sequence of points whose affine span is the space. The potential applications of these two results are considerable - whenever there is a natural spanning set for a vector space or affine space, computations can be done directly with it, in an efficient and stable way. This avoids the need to obtain a basis by thinning or applying an analogue of Gram-Schmidt to some ad hoc ordering of the spanning set, which may destroy the inherent geometry. Another typical situation where spanning sequences have natural advantages over bases is when the dimension of the space is difficult to determine, or sensitive to perturbations, as is the case for multivariate spline spaces. A number of indicative examples are given. These include vector spaces over the rationals, such as cyclotomic fields.

Keywords: finite frames, vector spaces over the rationals, least squares (minimum norm) solutions, affine spaces, barycentric coordinates, multivariate splines, cyclotomic fields

Math Review Classification: Primary 15A03, 15A21, 41A45, 42C15; Secondary 12Y05, 15B10, 41A15, 52B11, 65F25

Length: 21 pages

Last Updated: 10 May 2010