Fourier Analysis and Applications

Cod: 21161

Department: DCET

Department: DCET

ECTS: 6

Scientific area: Mathematics

Scientific area: Mathematics

Total working hours: 156

Total contact time: 26

Total contact time: 26

In this learning unit the student shall be introduced to Fourier Analysis and some of its applications. Starting by the definition and formal computations of Fourier series of givem functions, we proceed to the study of questions about pointwise, uniform, and mean quadratic (L^{2}) convergence. The applications will be to the solution of linear partial differential equations of Mathematical Physics and to the spectral analysis of stochastic processes.

Fourier Analysis

Partial differential equations

Stochastic processes

Partial differential equations

Stochastic processes

To understand and know how to apply Fourier Analysis to the study of classical linear partial differential equations of Mathematical Physics and to the spectral representation of stochastic processes and spectrum estimation.

1. Fourier series: definition and convergence notions and theorems

2. Classical partial differential equations of mathematical physics (heat, wave and Dirichlet): their solution via separation of variables and Fourier series

3. Analysis and spectral representation of stochastic processes; spectral density and autocovariance

4. Periodogram (significance tests), spectrum estimation and confidance intervals

2. Classical partial differential equations of mathematical physics (heat, wave and Dirichlet): their solution via separation of variables and Fourier series

3. Analysis and spectral representation of stochastic processes; spectral density and autocovariance

4. Periodogram (significance tests), spectrum estimation and confidance intervals

Bento Murteira, D. Muller & K. Turkman, *Análise de Sucessões Cronológicas*, McGraw-Hill , Lisboa, 2000.

Djairo Guedes de Figueiredo,*Análise de Fourier e Equações Diferenciais Parciais,* 2nd Edition, Projecto Euclides vol. 5, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1987.

Djairo Guedes de Figueiredo,

Online learning with continuous supervision favoring asynchronous communication (Moodle platform).

Students have to perform the tasks requested by the teacher: essays, critical recensions, reports, protocols, etc. All works will be evaluated and/or classified.

Students have to perform the tasks requested by the teacher: essays, critical recensions, reports, protocols, etc. All works will be evaluated and/or classified.

Continuous assessment is privileged: 2 or 3 digital written documents (e-folios) during the semester (40%) and a
presence-based final exam (p-folio) in the end of the semester (60%). In due time, students can alternatively choose to perform one
final presence-based exam (100%).

Students need to have solid knowledge of Mathematical Analysis at the undergraduate level, as well as working knowledge of ordinary differential equations and probability.