Abstract
We consider two classes of actions on ℝn—one continuous and one discrete. For matrices of the form A = e B with B ∈ M n(ℝ), we consider the action given by γ → γA t. We characterize the matrices A for which there is a cross-section for this action. The discrete action we consider is given by γ → γA k, where A ∈ GL n(ℝ). We characterize the matrices A for which there exists a cross-section for this action as well. We also characterize those A for which there exist special types of cross-sections; namely, bounded cross-sections and finite-measure cross-sections. Explicit examples of cross-sections are provided for each of the cases in which cross-sections exist. Finally, these explicit cross-sections are used to characterize those matrices for which there exist minimally supported frequency (MSF) wavelets with infinitely many wavelet functions. Along the way, we generalize a well-known aspect of the theory of shift-invariant spaces to shift-invariant spaces with infinitely many generators.
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Dedicated to John Benedetto.
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Larson, D., Schulz, E., Speegle, D., Taylor, K.F. (2006). Explicit Cross-Sections of Singly Generated Group Actions. In: Heil, C. (eds) Harmonic Analysis and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4504-7_10
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DOI: https://doi.org/10.1007/0-8176-4504-7_10
Publisher Name: Birkhäuser Boston
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