On the existence of Parseval frames for vector bundles

Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The <italic>Parseval frames</italic> are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles—this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors—and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.