Lectures log

• week 35: Review of basic material on the Fourier transform, see Groechenig Chapter 1. Heisenberg's uncertainty principle

with proof (Dym-McKean). Discussion of the extension of the Heisenberg uncertainty principle due to Robertson-Schroedinger, as discussed in Ricaud-Toressani, and its consequences for momentum and position operator, and for the wavelets, Lieb's inequalities wihthout proof

• week 36: Donoho-Stark's uncertainty principle (Groechenig, Chapter 2), Short-Time Fourier Transform, Donoho-Stark for STFT (Chapter 3 in Groechenig), wavelet transform
• week 37: Reproducing kernel Hilbert spaces (Chapter 1 & 2 in Paulsen)
• week 38: Bargman transform, Bargman-Fock space as reproducing kernel Hilbert space (partly Groechenig Chapter 4) Orthogonality relations for the STFT and the wavelet transform, reproducing kernel, reconstruction formula (Daubechies, Chapter 2)
• week 39: Bases in Banach and Hilbert spaces, biorthogonal systems (Christensen, Chapter 3) Gram matrix, Riesz bases in Hilbert spaces, frames in Hilbert spaces (Christensen, Chapter 3 & 5), Pseudo-inverse (Christensen Chapter 1 & 2)
• week 40: Basics on Gabor frames, Feichtinger's algebra (following Sections 1-3 in, M. S. Jakobsen: On a (no longer) new Segal algebra: a review of the Feichtinger algebra. Journal of Fourier Analysis and Applications 24 (6), 1579-1660, 2018), Janssen representation, Wexler-Raz biorthogonality principle, Density Theorem and Duality principle (M. S. Jakobsen, J. LemvigDensity and duality theorems for regular Gabor frames. Journal of Functional Analysis 270 (1), 229-263, 2016).
• week 41: Zak transform (following C. Heil: A Basis Theory.)