Lecture 1: Cardinal B-splines and convolution operators
Lecture 2: Poisson' summation formula and other prelims
Lecture 3: The analysis and synthesis operators, $L_2$ theory
Lecture 4: Frames
Lecture 5: Principal shift-invariant (PSI) space theory
Lecture 6: Linear independence in PSI spaces
Lecture 7: Wavelets: basics, quasi-affine systems and extension principles.
Lecture 8: wavelets: smoothness analysis. CAP systems.

 

일시 : 매주 화 목 1:00 - 2:30 (11월 17일 부터) 장소 : KAIST 산경동 3221호 The goal is to provide a glimpse into the basic concerning {it wavelet expansions} or {it wavelet representations}. The attempt will be to describe some of the basics in this field via the approach of {it shift-invariant spaces}. In view of the above, there will be essentially three (closely related) topics discussed in th lectures. (i) Representation of elements in a Hilbert space: The analysis and synthesis operators. Riesz bases, frames, tight frames, dual systems, dual bases, the canonical dual system. (ii) Shift-invariant spaces (one dimension, local PSI theory only): orthogonal proejection, Fourier transform characterization, approximation orders, linear independence, factorization, dual systems. (iii) wavelet systems: definition, the Haar wavelet, the sinc (Shannon) wavelet. Multiresolution analysis. Mallat's constructions. Daubechies' wavelets. Bi-orthogonal systems. Transfer operator analysis: smoothness, Riesz bases. (iv) wavelet frames: dual Gramian analysis of SI systems. Quasi-affine systems. The characterization of wavelet frames. Tight wavelet frames from multiresolution. Extension principles. Compactly supported tight spline frames. CAP/CAMP/LCAMP representations.