Aperiodic Coherent Frames Based on Primitive Tilings

• Main Author: Silvestre, Luis Jr.
• Format: text
• Published: Archīum Ateneo 2021
• Subjects: coherent frame, Lie n-group, coherent representation, primitive G-tiling, Q-frame
• Online Access: https://archium.ateneo.edu/theses-dissertations/538
• Institution: Ateneo De Manila University

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.