Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold
Keywords:Stiefel manifold, continuous frame, path-connected space, topological closure, dense subset
If H is a Hilbert space, the non-compact Stiefel manifold St(n, H) consists of independent n-tuples in H. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that ∩j∈J (U(j) + St(n, H)) is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating J-family. Then, in the second part, we show that the topological closure of St(n, H)∩S contains all polynomial paths contained in S and passing through a point in St(n, H). As a consequence, we prove that St(n, H) is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since St(n, L2(X, μ; F)) is isometric to, FF(X, μ), n, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.
D. Agrawal, The complete structure of linear and nonlinear deformations of frames on a Hilbert space. Master thesis, 2016.
D. Agrawal and J. Knisley, “Fiber Bundles and Parseval Frames”, 2015. arXiv:1512.03989
R. Balan, P. G. Casazza, C. Heil and Z. Landau, “Density, Overcompleteness, and Localization of Frames. I. Theory”, Journal of Fourier Analysis and Applications, vol. 12, pp. 105-143, 2006. https://doi.org/10.1007/s00041-006-6022-0
R. Balan, P. G. Casazza, C. Heil and Z. Landau, “Density, Overcompleteness, and Localization of Frames. II. Gabor Systems”, Journal of Fourier Analysis and Applications, vol. 12, pp. 307-344, 2006. https://doi.org/10.1007/s00041-005-5035-4
E. Bardelli and A. C. G. Mennucci, “Probability measures on infinite-dimensional Stiefel manifolds”, Journal of Geometric Mechanics, vol. 9, pp. 291-316, 2017. https://doi.org/10.3934/jgm.2017012
M. Bownik, “Connectivity and density in the set of framelets”, Mathematical Research Letters, vol. 14, no. 2, pp. 285-293, 2017. https://doi.org/10.4310/MRL.2007.v14.n2.a10
J. Cahill, D. Mixon and N. Strawn, “Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames”, SIAM Journal on Applied Algebra and Geometry, vol. 1, no. 1, pp. 38-72, 2017. https://doi.org/10.48550/arXiv.1311.4748
P. G. Casazza, “The art of frame theory”, Taiwanese Journal of Mathematics, vol. 4, no. 2, pp. 129-201, 2000. https://doi.org/10.11650/twjm/1500407227
P. G. Casazza and G. Kutyniok (eds.). Finite frames, theory and applications. Applied and Numerical Harmonic Analysis. New York: Birkhäuser, 2013.
O. Christensen, An Introduction to frames and Riesz Bases. 2nd ed. Switzerland: Birkhäuser, 2016.
O. Christensen, B. Deng and C. Heil, “Density of Gabor Frames”, Applied and Computational Harmonic Analysis, vol. 7, pp. 292-304, 1999. https://doi.org/10.1006/acha.1999.0271
I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions”, Journal of Mathematical Physics, vol. 27, pp. 1271-1283, 1986. https://doi.org/10.1063/1.527388
R. Duffin and A. Schaeffer, ”A class of non-harmonic Fourier series”, Transactions of the American Mathematical Society, vol. 72, pp. 341-366, 1952. https://doi.org/10.2307/1990760
K. Dykema and N. Strawn, “Manifold structure of spaces of spherical tight frames”, International Journal of Pure and Applied Mathematics, vol. 28, no. 2, pp. 217-256, 2006.
M. Frank and D. R. Larson, “Frames in Hilbert C*-Modules and C*-Algebras”, Journal of Operator Theory, vol. 48, no. 2, pp. 273-314, 2002.
D. Gabor, “Theory of communications”, Journal of the Institution of Electrical Engineers, vol. 93, pp. 429-457, 1946.
G. Garrigos, E. Hernandez, H. iki, F. Soria, G. Weiss, and E. Wilson, “Connectivity in the set of Tight Frame Wavelets (TFW)”, Glasnik matematicki, vol. 38, no. 1, 2003.
D. Han and D. R. Larson, “On the Orthogonality of Frames and the Density and Connectivity of Wavelet Frames”, Acta Applicandae Mathematicae, vol. 107, pp. 211-222, 2009. https://doi.org/10.1007/s10440-008-9420-2
P. Harms and A. C. G. Mennucci, “Geodesics in infinite dimensional Stiefel and Grassmann manifolds”, Comptes Rendus Mathematique, vol. 350, nos. 15-16, pp. 773-776, 2012. https://doi.org/10.1016/j.crma.2012.08.010
A. Hatcher, Algebraic topology. Cambridge University Press, 2005.
C. Heil, “History and Evolution of the Density Theorem for Gabor Frames”, Journal of Fourier Analysis and Applications, vol. 13, pp. 113-166, 2007. https://doi.org/10.1007/s00041-006-6073-2
O. Henkel, “Sphere-packing bounds in the Grassmann and Stiefel manifolds”, IEEE Transactions on Information Theory, vol. 51, no. 10, pp. 3445-3456, 2005. https://doi.org/10.1109/TIT.2005.855594
D. Husemoller, Fibre Bundles. 3rd ed. New York: Springer-Verlag, 1994.
I. M. James, The topology of Stiefel manifolds. London Mathematical Society Lecture Note Series. Cambridge: University Press, 1976.
V. Jurdjevic, I. Markina, and F. Silva Leite, “Extremal Curves on Stiefel and Grassmann Manifolds”, The Journal of Geometric Analysis, vol. 30, pp. 3948-3978, 2020. https://doi.org/10.1007/s12220-019-00223-1
D. Labate and E. Wilson, “Connectivity in the set of Gabor frames”, Applied and Computational Harmonic Analysis, vol. 18, no. 1, pp. 123-136, 2005. https://doi.org/10.1016/j.acha.2004.09.003
T. Needham and C. Shonkwiler, “Symplectic geometry and connectivity of spaces of frames”, 2019. arXiv:1804.05899v2
D. M. Speegle, “The s-elementary wavelets are path-connected”, Proceedings of the American Mathematical Society, vol.127, pp. 223-233, 1999.
N. Strawn, “Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations”, Journal of Fourier Analysis and Applications, vol. 17, pp. 821-853, 2011. https://doi.org/10.1007/s00041-010-9164-z
N. Strawn, Geometric structures and optimization on spaces of finite frames. PhD thesis, 2011.
N. Strawn, Geometry and constructions of finite frames. Master thesis, 2007.
S. F. D Waldron, An introduction to finite tight frames. Applied and Numerical Harmonic Analysis. New York: Birkhäuser, 2018.
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