Single-Course English 5 ECTS

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

See course in the course database.

Registration

Language

English

Duration

13 weeks

Institute

Compute

Place

DTU Lyngby Campus

Course code 01325
Course type Candidate
Semester start Week 5
Semester end Week 19
Days Wed 13-17
Price

7.500,00 DKK

Registration