Aperiodic Coherent Frames Based on Primitive Tilings

Date of Award


Document Type


Degree Name

Doctor of Philosophy in Mathematics



First Advisor

Job A. Nable, PhD


This thesis investigates aperiodic coherent frames for separable Hilbert spaces. Given a Lie n-group G with coherent representation (π, H) on a Hilbert space H, conditions on subsets ΛG ⊂ G, subsets F ⊂ H, transformations UG on G, and operators UH on H upon which F (UH(F), UG(ΛG)) = {π(λ)f : f ∈ UH(F), λ ∈ UG(ΛG)} forms a coherent frame for H are formulated for characterization. In particular, if a primitive G- tiling ΛG based on a primitive substitution G-tiling system (P, ω) and some F = {φ} accordingly satisfy such requirements, this study proves that F ({UH(φ)} ,Λ) is an ape- riodic frame for H under some appropriate operator UH on H for every Λ in the continu- ous hull of ΛG. Specific cases when G is compact, G = U(H), and H = L 2 (R d ) are ex- amined for specialized results. Lastly, Q-frames and discretized localization operators are introduced to establish significance of coherent frames in Physics and demonstrate its physical applications.

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