Math 414 Course Outline

I. Motivation - Data Compression and filtering

II. Background

(Roughly 3 weeks)

• Inner Product Spaces
• L^2 Spaces

III. Fourier Series

(Roughly 3 weeks)

• Computation of Fourier Series
• Convergence of Fourier Series

IV. Fourier Transform

(Roughly 3 weeks)

• Definitions
• Basic Properties
• Inverse Fourier Transform
• Sampling Theorem
• Uncertainty Principle
• Discrete Fourier Transform

V. Wavelets

(Roughly 4 weeks)

• Haar Wavelets
• Multiresolution Analysis
• Decompostion and Reconstruction
• Wavelets and the Fourier Transform
• Daubechies' Wavelets

VI. Further Topics

(As time permits)

• Wavelet Transform
• Quadrature Mirror Filters