Admission requirements
Knowledge of linear algebra, elementary topology, real analysis, Lebesgue integration.
It is highly recommended that the students have followed the courses Inleiding Maattheorie and Linear Analysis.
Knowledge of complex analysis is an advantage, but it is not required.
Description
The course focuses on the theory and applications of Fourier series and the Fourier transform.
In the first part of the course we will introduce classical Fourier theory, focusing on the Fourier series for periodic functions.
In the second part of the course we will describe the Fourier transform on the real line, culminating in
the Fourier transform for test functions and (tempered) distributions. If time allows, we will highlight some applications to other areas of mathematics.
Course objectives
At the end of the course, the student knows and understands the treated course material and is able to reproduce the content of the course. The student is able to apply the methods of Fourier analysis to concrete problems and understands to problems from different areas of mathematics.
Timetable
You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.
MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).
For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.
Mode of instruction
Lectures: 2 hours per week.
During the course, individual students can hand in up to six assignments, according to a schedule that is provided in Brightspace at the start of the semester.
Assessment method
During the course, individual students can hand in a number of assignments, according to a schedule that is provided in Blackboard at the start of the semester. The lowest grade for these is stricken, and the homework grade is determined as the (unrounded) average of the remaining grades for these assignments.
The unrounded final grade for the course is the maximum of:
the weighted average of the homework grade (25%),
the maximum of the (unrounded) grades for the written exam and the resit (75%),
the (unrounded) grade for the written exam, and
the (unrounded) grade for the resit.
This maximum is then rounded to the nearest half-integer, but not to 5.5, to obtain the final grade for the course. If the result is 6.0 or higher, this is a "pass", provided that a grade of at least 5.0 (unrounded) has been obtained for the written exam and/or the resit. If the result is 5.0 or lower, or if no grade of at least 5.0 (unrounded) has been obtained for the written exam and/or the resit, this is a "fail".
During the written exam and the resit you can use the textbook of the course and the notes that you may have taken.
Reading list
Grafakos, Loukas. Classical Fourier Analysis. Vol. 249. New York, NY: Springer, 2014
Stein, Elias M., and Rami. Shakarchi. Fourier Analysis : an Introduction. Princeton, NJ: Princeton University Press, 2003.
Stein, Elias M., and Rami. Shakarchi. Real Analysis : Measure Theory, Integration, and Hilbert Spaces. Princeton, NJ ; London: Princeton University Press, 2005.
Vretblad, Anders. Fourier Analysis and Its Applications. New York: Springer, 2003.
Registration
From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.
Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.
Extensive FAQ's on MyStudymap can be found here.
Contact
Both the instructor and the assistants can be reached at the dedicated email address fourieranalysis@math.leidenuniv.nl