Syllabus

Module I (8 hours)
Review of Fourier series, Fourier theory on L1 (R) and L2 (R), basic properties of windowed Fourier transform.

Module II (4 hours)
Orthonormal basis generated by a single function, Balian-Low theorem, Gabor transform.

Module III (18 hours)
Dyadic wavelets and examples, Multiresolution Analysis (MRA), scaling function and low-pass filters, construction of orthonormal wavelets from MRA, Properties of MRA wavelets, Properties/characterizations of scaling functions and low-pass filters, Spline wavelets, Daubechies wavelets, Some applications of wavelets.

To identify the limitations of classical Fourier analysis.

To introduce the construction/applications of wavelets with some specific examples.

CO1: Students will learn about results related to convergence, term-by-term integration and differentiation of Fourier series along with some applications.

CO2. They will learn about Fourier transform on L1(R) and its extension to L2(R) along with their properties.

CO3. Students will get acquainted with the properties and characterizations of wavelets in L2(R) and they will learn about the powerful technique of MRA for the construction of wavelets.

CO4. They will see the construction of different types of wavelets such as Daubechies and Spline wavelets.

CO5. Also they will get familiar with some applications of wavelets in signal/image processing.

E. Hernandeze and G. Weiss, A first course on wavelets, CRC Press

D. F. Walnut, An introduction to wavelet analysis, Birkhäuser

G. Bachman, L. Narici, and E. Beckensterin, Fourier and Wavelet Analysis, Springer-Verlag

C.K . Chui, An Introduction to Wavelets, Academic Press