# Frames for vector spaces and affine spaces

## Shayne Waldron

## Abstract:

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

## Availability: