Syllabus Information

1.1. Introduction

This is a six-credit, optional course in the degree of Mathematics. Its aim is to present the most important concepts and properties of Fourier series, for periodic functions, and Fourier transforms, for non-periodic functions in one variable, together with some elementary applications of the theory to signal processing.

1.2. Recommendations to take this course

The attendace to the class lectures and the computer laboratory sessions is advisable, as well as the individual work on the problems posed along the course and the use of the individual tutorization.

It is highly advisable to have passed the module Iniciación al Análisis matemático, which comprises the subjects Análisis matemático I, Análisis matemático II and Variable compleja. The subject requires a good knowledge of the Lebesgue integral and the Lebesgue spaces L¹ and L².

1.3. Context and importance of this course in the degree

The subject belongs to the module Ampliación de análisis matemático. To take this subject it is advisable to have passed the subjects of the module Iniciación al análisis matemático, that is, Análisis matemático I, Análisis matemático II and Variable compleja, as well as the subject Integral de Lebesgue. The subject Análisis funcional is a good companion.

1.4. Activities and key dates

Problems or homework tasks will be posed during the course, allowing to pass the subject.

There will be a final exam of the subject during the oficial period set by the Faculty of Science.

2.1. Learning goals

At the end of this course students should be able to:

• Know that a periodic function is determined by its Fourier coefficients and understand some convergence results of Fourier series.
• Know how the Fourier coefficients can be obtained by the discrete Fourier transform, and use the basics of the fast Fourier transform.
• Adapt the theory to non-periodic functions with the Fourier transform and understand some inversion results.

3.1. Aims of the course

This is an optional course in the degree of Mathematics. Its goal is to present the essentials of Fourier analysis.

4.1. Assessment tasks (description of tasks, marking system and assessment criteria)

As a general rule, the module can be passed either showing a regular work along the academic year, or by a final exam.

• Regular work. During the course, the student results will be evaluated through a periodical supply of exercises or short tasks, together with their active participation during the course. The use of LaTeX in written presentations is recommended; the evaluation can also include oral presentations. These evaluations will constitute the final mark.
• Final exam. The aforementioned procedure does not exclude the right, according to the current regulations, to a final exam which, by itself, allows to pass the module.

5.1. Methodological overview

The learning process designed for this course is based in the following:

• Explanation of the theoretical contents of the subject in class lectures.
• Practical application of the theoretical results by software tools in computer laboratories.
• Individual work on problems and homework assignments.

5.2. Learning tasks

• Class lectures for explaining the theoretical results.
• Computer laboratory sessions using software tools to illustrate applications of the theoretical results seen in class lectures.
• Individual problem and homework task assignment.
• Individual tutorization.
• Use of the computer tools of the University, mainly the computer laboratories and the Anillo digital docente.

5.3. Syllabus

• Historical, physical and mathematical introduction. The vibrating string and the wave equation: D'Alembert, Euler, and Bernoulli. The heat transmission and its equation: Fourier. The concept of function: measure theory and functional analysis. The electromagnetic waves.
• Preliminary mathematics. Banach spaces of continuous, differentiable, and integrable functions. Convergence of sequences and series of functions. Periodic functions, the torus and a little bit of complex analysis.
• Fourier series. Formal Fourier sine, cosine and exponential series. Statement of the problem of the convergence of Fourier series: convolution, kernels, the unit circle, and its relation with complex analysis and the involved spaces. Pointwise, uniform and mean convergence results: summabilities of Fourier series. Riemann-Lebesgue lemma. Dirichlet's theorem and the Gibbs phenomenon. Riemann's localization principle. Exploiting the orthogonality: Hilbert spaces and Plancherel's theorem.
• Discrete Fourier transform. Periodic sequences. The discrete transform and its inverse. Sampling and interpolation. Approximate calculus of Fourier coefficients. The FFT algorithm and its use in computer programs (Python).
• Fourier transform. The continuous analog of Fourier series. Continuous frequencies. The Schwartz class of functions. Poisson and Gauss-Weierstrass kernels. The inversion formula. Fourier transform and L² theory. Band limited functions. The uncertainty principle.

5.4. Course planning and calendar

• As a general rule, this subject has four presential hours per week. The schedule is set and made public by the Faculty of Science well before the beginning of the academic year.
• The computer laboratory sessions will be scheduled according to the general calendar in coordination with the class lectures.
• Problems or homework tasks, and their deadlines, will be posed in coordination with the class lectures.
• The dates of the exams are set and made public by the Faculty of Science well in advance before the beginning of the academic year.

5.5. Bibliography and recommended resources

• F. J. Ruiz Blasco: Análisis de Fourier (in Spanish). Available for students in the Anillo digital docente of the University of Zaragoza.
• E. M. Stein and R. Shakarchi: Fourier analysis. An introduction. Princeton Lectures in Analysis, 1. Princeton University Press, Princeton, NJ, 2003. xvi+311 pp. ISBN: 0-691-11384-X.
• J. Duoandikoetxea: Fourier analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. xviii+222 pp. ISBN: 0-8218-2172-5.
• Y. Katznelson: An introduction to harmonic analysis. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004. xviii+314 pp. ISBN: 0-521-83829-0; 0-521-54359-2.
• W. Rudin: Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1.