**MTH ****521: Introduction to Wavelets**** ****(4)**

**Pre-requisites (Desirable)**: MTH 311, MTH 404

*Learning Objectives*:

This is an introductory course on wavelet analysis. In this course we will introduce the basic notion of wavelets in different settings, namely for finite groups, discrete infinite groups and real line. This will provide the students an opportunity to know perspective applications of linear algebra and real analysis in mathematics and beyond.

*Course Contents*:

**Review of Linear Algebra**: Complex Series, Euler’s Formula, Roots of Unity, Linear Transformations and Matrices, Change of Basis, diagonalization of Linear Transformations and Matrices, Inner Product, Orthogonal Bases, Unitary Matrices.

**The Discrete Fourier Transform**: Definition and Basic Properties of Discrete Fourier Transform, Translation-Invariant Linear Transformations, The Fast Fourier Transform.

**Wavelets on Finite Group Z _{N}**: Convolution on Z

_{N}, Fourier Transform on Z

_{N}, Definition of Wavelets and Basic Properties, Construction of Wavelets on Z

_{N}.

**Wavelets on Infinite Discrete Group Z** : Definition and Basic Properties of Hilbert spaces, Complete orthonormal Sets in Hilbert Spaces, The spaces l_{2}(Z) and L^{2}([-π, π)), Basic Fourier Series, The Fourier Transform and Convolution on l_{2}(Z) Wavelets on Z.

**Wavelets on R**: Convolution and Approximate Identities, Fourier Transform on R, Bases for The Space L^{2}(R), Belian-Low Theorem, Wavelets on R, Multiresolution Analysis, Construction of Wavelets from multiresolution Analysis, Construction of Compactly supported Wavelets, Haar Wavelets, Band-Limited Wavelets, Applications.

*Suggested Books*:

- Michael W. Frazier: An Introduction to Wavelets Through Linear Algebra, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1999.
- Eugenio Hernandez, Guido Weiss: A First Course on Wavelets, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1996.
- Ingrid, Daubechies: Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992

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