Mathematics 4: Analysis – a Toolbox in Physics and Engineering

• 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
• apply precise mathematical arguments and calculations concerning Hilbert spaces and their operators, e.g., within physics and quantum mechanics
• 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
• apply B-splines in mathematical and application-oriented contexts
• understand the role of reproducing kernel Hilbert spaces in machine learning

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, B-splines, reproducing kernel Hilbert spaces

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.

Lectures and exercises