Topic outline

  • General

    Linear analysis, 7.5 hp

    Teaching. Lectures and lessons on Tuesdays and Fridays at 12.45-15.30 in room 32, house 5, Kräftriket (Roslagsvägen 101), see schedule. Note that the teaching on Tuesday, November 18 has been moved to Wednesday, November 26.

    Course literature. Allan Pinkus and Samy Zafrany, Fourier Series and Integral Transforms, Cambridge University Press, 1997, ISBN: 9780521597715. The book is available at Bokus. Lecture notes on those parts of the course that cannot be found in the book by Pinkus and Zafrany (will be published on the course home page next week).

    Teacher. Thomas Önskog thomas.onskog@math.su.se.

    Course contents. The course treats Fourier series with applications to boundary value problems. Also other topics within modern Fourier analysis is covered. The contents of the course can be used for modelling in for example physics.

    Written exam. The course is examined by means of a mandatory written exam on January 17 (Saturday) at 9.00-14.00. The exam consists of 6 problems, each worth 5 points. On each of the problems, a certain fraction of the five points will be awarded for calculations (below denoted C) and a certain fraction for theoretical arguments and written presentation (below denoted T). The number of possible C- and T-points will vary from one problem to another. The grades F, FX, E, D and C will be based on the total number of C- and T-points according to the grading criteria on the course home page.

    Oral exam. Students that score above a certain level at the written exam will be granted the possibility to take an optional oral exam. Passing the oral exam will be mandatory for grades A and B.

    Bonus: A bonus of at most two C-points and one T-point can be obtained at the written exam in the following way: Hand in solutions to the problems suggested at the home page (marked with B below) no later than one week after the lecture in question. These solutions are then corrected and awarded either 1C or 0C (depending on the quality of the calculations) and either 1T och 0T (depending on the quality of the reasoning). To obtain the C-bonus, at least two thirds of the possible C-points must have been obtained and to obtain the T-bonus at least two thirds of the possible T-points must have been obtained.

  • Day 1, November 4

    Motivation and introduction to Fourier series. Review of vector spaces and inner product spaces.

    Text: PZ: 1.1, LW: 1.

    Exercises: 1.1 (1B, 2, 3, 4, 5B, 6).

  • Day 2, November 7

    Inner product spaces. Orthogonal systems

    Text: PZ: 1.2–1.4.

    Exercises : 1.2 (1B, 2, 6, 8), 1.3 (2), 1.4 (2B, 5).

    • Day 3, November 11

      Inifinite orthonormal systems. Fourier series

      Text: PZ: 1.5, 2.1.

      Exercises: 1.5 (2, 4B, 5), 2.1 (1B,4,5).

      • Day 4, November 14

        Pointwise convergence of Fourier series

        Text: PZ: 2.2–2.4.

        Exercises: 2.3 (1, 4B, 5), 2.4 (1, 3B, 8).

      • Day 5, November 21

        Pointwise convergence of Fourier series (cont'd). Uniform convergence.

        Text: PZ: 2.4–2.5.

        Exercises : 2.4 (2B, 4, 7), 2.5 (1, 4B, 5).

        • Day 6, November 25

          Parseval's identity. Gibbs' phenomenon

          Text: PZ: 2.6–2.7.

          Exercises: 2.6 (1B, 3), AE (1, 2B).

          • Day 7, November 26

            Sine and cosine series. Differential and integration of Fourier series. Fourier series on other intervals.

            Text: PZ: 2.8–2.10.

            Exercises: 2.8 (1, 2, 3B), 2.9 (1B, 3, 7).    
                    

          • Day 8, November 28

            Applications of Fourier series

            Text: LW: 2.

            Exercises: 2.10 (2B, 3, 4), Review exercises to chapter 2 (3, 7B, 8).

          • Day 9, December 2

            Sturm-Liouville problems and applications to partial differential equations.

            Text: LW: 3.1.

            Exercises: AE (3, 4B, 5B, 6).

            • Day 10, December 5

              Sturm-Liouville problems (cont'd). Boundary value problems for partial differential equations.

              Text: LW: 3.1 (Ex. 3.9), 3.2 and 3.3 (Case 1).

              Exercises: AE (7B, 8, 9B).

              • Day 11, December 9

                Boundary value problems for partial differential equations (cont'd). Wavelets.

                Text: LW: 3.3 (Cases 2 and 3), 4.1.

                Exercises: AE (10B, 11B, 12).

                • Day 12, December 12

                  Wavelets (cont'd). Fourier transform.

                  Text: LW: 4.2, PZ: 3.1–3.3.

                  Exercises: AE (13B, 14, 15, 16B).

                  • Day 13, December 16

                    Fourier transform (cont'd).

                    Text: PZ: 3.3–3.5, 3.7.

                    Exercises: 3.2 (1aeh), 3.3 (1, 3, 5B, 6, 7), 3.4 (1, 2a, 4B).

                    • Day 14, December 19

                      Signal processing. Discrete Fourier transform.

                      Text: PZ: 3.8, O13.

                      Exercises: 3.4 (5B), 3.5 (1B, 5).

                      • Day 15, January 9

                        Repetition.