Groups are selected in myStudies. Mi 12-14 als Ausweichtermin für jene, welche Mi 10-12 das Wahlpflichtfach "Introduction to Graph Theory" besuchen. Some of the exercise classes will be taught in English.

This class covers the basic theory of Hilbert spaces, Fourier series and Fourier Transform, and its application to the study of classical linear PDEs.

Objective

1) Learn the basic theory of Hilbert spaces, Fourier series, Fourier Transform. Understand the strong physical intuition behind these mathematical concepts. 2) Learn about some concrete problems that can be effectively attacked with these tools, and understand what is the rigorous interpretation of the abstract results in such problems. Get a feeling about how to recognize such problems. 3) Learn what are the typical limitations and shortcomings of these tools.

Content

1) Real and complex Hilbert spaces, Hilbert bases and Riesz representation Theorem 2) Fourier series of a function in L^2([-π, π]; C), relationship between the regularity of a function and the asymptotic behaviour of the Fourier coefficients. Application to the resolution of linear partial differential equations with various boundary conditions in [-π, π]. 3) Fourier Transform in R^d and its elementary properties, relationship between the regularity of the function and the asymptotic behaviour of its Fourier transform, relationship between the summability of the function and the regularity of it Fourier transform. Application to the resolution of linear partial differential equations with various decay conditions in R^d. 4) Compact operators on Hilbert spaces, Self-adjoint operators, the spectral theorem, eigenvalue problems, and applications.

Lecture notes

Notes (typesetted or handwritten) will be made available as they are produced to enroled students.

Literature

The course will not follow a specific text, hence live participation is recommended. The material can be found in - Fourier Analysis : An Introduction, E. Stein, R. Shakarchi

Prerequisites / Notice

Linear Algebra, Analysis I, II and III (Mass and integration)

Competencies

Subject-specific Competencies

Concepts and Theories

assessed

Techniques and Technologies

assessed

Method-specific Competencies

Problem-solving

assessed

Personal Competencies

Creative Thinking

assessed

Critical Thinking

assessed

Performance assessment

Performance assessment information (valid until the course unit is held again)

Performance assessment as a two-semester course together with 401-2283-00L Analysis III (Measure Theory)

For programme regulations
(Examination block)

Bachelor's Degree Programme in Mathematics 2021; Version 07.04.2022 (Examination Block 2)

The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.

Mode of examination

oral 20 minutes

If the course unit is part of an examination block, the credits are allocated for the successful completion of the whole block. This information can be updated until the beginning of the semester; information on the examination timetable is binding.