In this curricular unit several aspects of Harmonic Analysis, on a beginning postgraduate level, are considered, and results and proofs are discussed on pointwise, uniform, and mean square convergence of Fourier series, as well as the summation methods of Abel and Cesàro. The discrete Fourier and Haar transforms (including FFT and FHT) are studied, as well as the Fourier transform in S and S'. The curricular unit concludes with an introductory study to Wavelet Analysis and its applications. The goal of this learning trajectory is to introduce the student to modern methods of Applied Harmonic Analysis in a relatively straightforward way, and to provide him/her with the conceptual and technical instruments that will allow him/her to understand the recent scientific literature and to proceed more advanced post-graduate studies in Applied Analysis or in other scientific and technological areas that need those instruments.
Fourier series Discrete Fourier and Haar transforms Wavelets Harmonic Analysis
Upon completion of this curricular unit the student should be able to recognize the concepts of harmonic analysis and know how apply the methods of harmonic analysis in the resolution of problems.
1. Fourier series: motivation, introduction, historical notes
2. Pointwise convergence of Fourier series
3. Summation methods (Abel and Cesàro)
4. Mean square convergence
5. Discrete Fourier and Haar transforms (including FFT and FHT)
6. Fourier transforms in S and in S'
8. Multiresolution analysis
9. Applications of wavelet analysis
Albert Boggess, Francis J. Narcowich: A First Course in Wavelets with Fourier Analysis, 2nd Edition, Wiley. Secundary Reading:
M.C. Pereyra, L.A. Ward, Harmonic Analysis: From Fourier to Wavelets, Student Mathematical Library IAS/Park City, Mathematical Subseries, volume 63, American Mathematical Society/Institute for Advanced Study, Providence RI/Princeton NJ, 2012.
E.M. Stein, R. Shakarchi, Fourier Analysis: an introduction, Princeton Lectures in Analysis I, Princeton UniversityPress, Princeton NJ, 2003.
D. G. de Figueiredo: Análise de Fourier e Equações Diferenciais Parciais, Projecto Euclides vol. 5, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1987.
Evaluation is made on individual basis and it involves the coexistence of two modes: continuous assessment (60%) and
final evaluation (40%). Further information is detailed in the Learning Agreement of the course unit.
The prerequisites for this CU are a thorough knowledge of Linear Algebra and Infinitesimal Analysis.