Journal of Fourier Analysis and Applications

, Volume 17, Issue 5, pp 821–853 | Cite as

Finite Frame Varieties: Nonsingular Points, Tangent Spaces, and Explicit Local Parameterizations



The (μ,S)-frames are frames with lengths in [μ 1⋅⋅⋅μ N ] and with frame operator S, or the \(F=[f_{1}\cdots f_{N}]\in M_{d\times N}(\mathbb{E})\) with column lengths listed by μ and which satisfy FF =S. In this paper, we characterize the nonsingular points of real and complex finite (μ,S)-frame varieties by determining where generalized tori and distorted Stiefel manifolds intersect transversally in Hilbert-Schmidt spheres. This provides us with a characterization of the tangent space for each nonsingular point of the (μ,S)-frame varieties, and we leverage this characterization to demonstrate the existence of structured, locally well defined analytic coordinate patches. We conclude by deriving explicit expressions for these coordinates.


Finite frames FUNTF Local parameterizations 

Mathematics Subject Classification (2000)

14M99 15B99 47B99 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

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