  Parseval frames can be thought of as redundant or linearly dependent
coordinate systems for Hilbert spaces, and have important applications in such
areas as signal processing, data compression, and sampling theory. We extend
the notion of a Parseval frame for a fixed Hilbert space to that of a moving
Parseval frame for a vector bundle over a manifold. Many vector bundles do not
have a moving basis, but in contrast to this every vector bundle over a
paracompact manifold has a moving Parseval frame. We prove that a sequence of
sections of a vector bundle is a moving Parseval frame if and only if the
sections are the orthogonal projection of a moving orthonormal basis for a
larger vector bundle. In the case that our vector bundle is the tangent bundle
of a Riemannian manifold, we prove that a sequence of vector fields is a
Parseval frame for the tangent bundle of a Riemannian manifold if and only if
the vector fields are the orthogonal projection of a moving orthonormal basis
for the tangent bundle of a larger Riemannian manifold.

Geometric Topology;Functional Analysis