2012/09/25 by Daniel Freeman, Freeman, Daniel, Ryan Hotovy +3
Earth and Planetary Sciences · Engineering · Mathematics · #42C15 #57R22 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Topology (math.GT) #Mathematical Analysis and Transform Methods #Seismic Imaging and Inversion Techniques #math.FA #math.GT #msc:42C15 #msc:57R22
paper · pdf · doi:10.48550/arxiv.1209.5495
15 pages
arxiv created 2012/09/25 · openalex publication_date 2012/09/25 · arxiv updated 2012/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Frames for \Rn can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point which varies smoothly over the manifold. It is well known that the only spheres with a moving basis for their tangent bundle are S1, S3, and S7. On the other hand, after combining the two separate meanings of the word "frame", we show that the n-dimensional sphere, Sn, has a moving finite unit tight frame for its tangent bundle if and only if n is odd. We give a procedure for creating vector fields on S2n-1 for all n∈\N, and we characterize exactly when sets of such vector fields form a moving finite unit tight frame.