Function spaces and mathematical analysis
Overall Course Objectives
To provide the students with the mathematical background that is needed for studies in physics and applied mathematics
Learning Objectives
- 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
- master basic wavelet theory
- perform calculations with the L^p-spaces and the corresponding sequence spaces
Course Content
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, Sturm-Liouville theory and special basis functions (e.g, Legendre and Hermite polynomials), an introduction to wavelet theory. Additional topics can include: B-splines, Finite Element Method, etc.
Recommended prerequisites
01035/01025/01034/01037, [Subjects from 01001.01002/01005, 01020, or an equivalent course:] Linear equations and linear maps. Matrix algebra. Vector spaces. Eigenvalue problems. Symmetric and orthogonal matrices. Complex numbers. Linear differential equations. Standard functions. Functions of one and several real variables: linear approximations and partial derivatives, Taylor expansions, and quadratic forms.
[Subjects from 01035 or an equivalent course:] Infinite series, power series, Fourier series. Convergence (absolute, conditional, point-wise, uniform) of infinite series, Introduction to the Fourier transform.
Teaching Method
Lectures and exercises