Advanced Real Analysis with Ole Christensen

Course Description

This is a Master's graduate-level course on real analysis. A student who has met the objectives of the course will be able to:
- distinguish between normed spaces and Hilbert spaces
- understand various types of convergence and how to verify them
- master basic operations in Hilbert spaces
- understand the role of linear algebra in analysis
- know the role of L^2 and perform basic operations herein
- master the basic manipulations with Fourier transform
- know when one should apply Fourier series or the Fourier transform
- expand square-integrable functions in various bases
- Perform calculations on B-splines
- Perform calculations with the L^p-spaces and the corresponding sequence spaces
- master basic wavelet theory

Some of the topics covered include: Normed vector spaces, Hilbert spaces, bases in Hilbert spaces, basic operator theory, the spaces L^p and l^p, approximation, the Fourier transform, convolution, the sampling theorem, B-splines, special basis functions (e.g, Legendre and Hermite polynomials), an introduction to wavelet theory.