Frames for vector spaces and affine spaces |
| |
Authors: | Shayne Waldron |
| |
Institution: | Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand |
| |
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, and thus it is possible to extend the frame expansion (and other elements of frame theory) to any finite spanning sequence for a vector space. The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact that the associated Gramian matrix is an orthogonal projection. Existing generalisations of the frame expansion to Banach spaces involve an analogue of the frame bounds and frame operator.The potential applications of our results are considerable. Whenever there is a natural spanning set for a vector space, computations can be done directly with it, in an efficient and stable way. We illustrate this with a diverse range of examples, including multivariate spline spaces, generalised barycentric coordinates, and vector spaces over the rationals, such as the cyclotomic fields. |
| |
Keywords: | Primary: 15A03 15A21 41A45 42C15 Secondary: 12Y05 15B10 41A15 52B11 65F25 |
本文献已被 ScienceDirect 等数据库收录! |
|