Global coordinate systems: Continuously moving finite-dimensional unit norm tight frames on smooth manifolds
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Continuously moving bases for tangent spaces of manifolds are important in the study of differential geometry and mathematical physics. However, globally continuous bases do not exist for the tangent spaces of all manifolds, for instance the Mobius strip and the 2-dimensional sphere. Some applications, particularly those in signal processing, call for a more general coordinate system known as a tight frame. Finite-dimensional unit-norm tight frames (FUNTFs) are a natural generalization of an orthonormal basis which satisfy a useful reconstruction formula for all vectors in their span. Thus, we are motivated to study the existence of FUNTFs for the tangent spaces of manifolds. We investigate questions about the existence of FUNTFs on manifolds, the minimum number of vectors needed for a FUNTF, and potential applications. In particular, we study the Mobius strip, its higher dimensional generalization of vector bundles on the circle, and n-spheres.