Introduction

Course objectives / Syllabus / Assessment Methodology.

Vector spaces. Inner product on a vector space. Norm and distance measure in a vector space. Squareness. Convergence of a sequence in a vector space. Closure of a vector space. Closure of span of vectors.

Euclidean vector spaces (normalized projections, construction of bi -orthogonal bases, construction of frames)

Completeness of a vector space. Hilbert spaces. Separability of a Hilbert space. Orthogonal complement of a Hilbert space, the direct sum of vector spaces.

Definition of linear operator. Norm of an operator. Bounded linear operator. Kernel and image of a linear operator. Inverse operator. Examples of calculation of norms of a linear operator. Examples of an operator inverse. Adjoint operator. Examples of an adjoint operator. (Self-)adjoint operator. Existence and uniqueness of an adjoint operator. Unitary operator. Eigenvalue / eigenvector of an operator. (Semi-)positive definite operators.

Approximations, projections, decompositions - projection theorem; projection operators; direct sums and sub-space decomposition, least square estimation.

Bases and frames: main definition, Bessel's inequality. Equivalence theorem, least square approximation, the definition of bi-orthogonal bases ( Riesz bases ), frames, tight frames, 1-tight frames.

Complements of linear algebra: matrices, elementary matrices (I, J, diagonal, symmetric, hermitian); Operations on matrices (sum , product, Kronecker product, transposition, Hermitian transposition) and their properties; various definitions and associated properties : determinant, rank, inversion of a matrix, systems of linear equations (exact solutions, approximate, null space, range); Diagonalization of a matrix. Families of matrices (symmetric, circulant right and left, Toeplitz, positive definite, root of a positive definite matrix, rank of a parametric matrix, polynomial matrices, unimodular, paraunitary)

Time-frequency representation of "prototype" functions; spectral expansion/ temporal expansion of a mother function. Heisenberg 's uncertainty principle. Optimal time-frequency localization. Fourier transform. STFT. Gabor transform. Spectrogram associated with a STFT. Example of localization of the STFT. Time-frequency representation of a WT. Concept of scale / resolution (discrete-time versus continuous-time)

CWT : definition, inversion, properties (linearity, Shift-Invariance, Scale change). Scalograms.

CWT properties ( Conservation of energy and of the scalar product , reproducing Kernel); influence cone for a finite time signal support / influence region for a band-limited signal.

Morlet wavelet / other examples.

Over-sampling tight frame for reducing quantization error.

Sampling in time/in scale of the CWT. Perfect reconstruction constraint. Dyadic scaling .

Discrete wavelet expansion of continuous signals - Haar Wavelet: orthonormality of the basic functions and Haar expansion / Completeness of the Haar expansion for signals in L2(R). 2-channel filter bank for discrete signals - DWT.

Multi-resolution expansion. Iteration of a 2-channel filter bank to the WT mother function.

Orthonormal DWT. DWT bi-orthogonal and links with wavelet series representation of continuous signals.