Frame decompositions, sampling, and uncertainty principle inequalities
John J. Benedetto · 2021
Gabor’s approach to signal analysis has been formalized and generalized in several ways. The approach in this chapter integrates signal reconstruction and the uncertainty principle. The signal reconstruction is in terms of frame decompositions and sampling theory, with an emphasis on local sampling and effective irregular sampling theorems and algorithms. The uncertainty principle is formulated in terms of the uncertainty principle for variances in the context of Gabor and wavelet systems; and is developed to include general weighted uncertainty principle inequalities. Examples include dissections of the information plane, real-time sampling, auditory models, stationary systems, the Plancherel-Pólya and Boas theory, aliasing, zero-crossings, entropy inequalities, uniqueness theory and logarithmic integrals, local uncertainty principle inequalities, and the role of Riesz transforms and A p - spaces. The theory unifies some of these notions in the parlance of wavelets and coherent states. The methods are from harmonic and functional analysis.