# 401-2464-00L  Analysis IV (Fourier Theory and Hilbert Spaces)

 Semester Spring Semester 2024 Lecturers M. Iacobelli Periodicity yearly recurring course Language of instruction English (lecture), German (exercise)

### Courses

NumberTitleHoursLecturers
401-2464-00 VAnalysis IV (Fourier Theory and Hilbert Spaces)3 hrs
 Wed 09:15-10:00 HG F 3 » Fri 10:15-12:00 HG F 3 »
M. Iacobelli
401-2464-00 UAnalysis IV (Fouriertheorie und Hilberträume)
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.
2 hrs
 Wed 10:15-12:00 HG G 26.1 » 10:15-12:00 LEE D 105 » 10:15-12:00 ML F 40 » 10:15-12:00 ML H 43 » 10:15-12:00 ML J 34.1 » 12:15-14:00 ML F 40 »
M. Iacobelli

### Catalogue data

AbstractThis class covers the basic theory of Hilbert spaces, Fourier series and Fourier Transform, and its application to the study of classical linear PDEs.
Objective1) 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.
Content1) 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 notesNotes (typesetted or handwritten) will be made available as they are produced to enroled students.
LiteratureThe 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 / NoticeLinear 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) ECTS credits 12 credits Examiners F. Da Lio, M. Iacobelli Type session examination Language of examination English Repetition The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit. Mode of examination written 180 minutes Written aids None Performance assessment as a semester course (other programmes) ECTS credits 6 credits Examiners M. Iacobelli Type session examination Language of examination English Repetition 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.

### Learning materials

 Main link Information Only public learning materials are listed.

### Groups

401-2464-00 UAnalysis IV (Fouriertheorie und Hilberträume)
GroupsG-01
 Wed 10:15-12:00 HG G 26.1 »
G-02
 Wed 10:15-12:00 LEE D 105 »
G-03
 Wed 10:15-12:00 ML F 40 »
G-04
 Wed 10:15-12:00 ML H 43 »
G-05
 Wed 10:15-12:00 ML J 34.1 »
G-06
 Wed 12:15-14:00 ML F 40 »

### Restrictions

 There are no additional restrictions for the registration.

### Offered in

ProgrammeSectionType
Mathematics BachelorExamination Block 2O