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Fourier Analysis

Code 15618
Year 1
Semester S2
ECTS Credits 8
Workload TP(60H)
Scientific area Mathematics
Entry requirements Does not have.
Learning outcomes The aim of this course is to give a rigorous introduction to Fourier series, Fourier transform and discrete Fourier transform, including some applications.

At the end of this curricular unit, thebstudents should:
i. know the fundamental aspects of the theory of Fourier series, Fourier transforms and discrete Fourier transforms;
ii. be able to apply definitions and central results of Fourier Analysis in different contexts, in particular in the study of equations with partial derivatives and in Number Theory;
iii. be able to mobilize knowledge of Fourier Analysis to address problems in different areas of Science and Technology.
Syllabus 1. Fourier series
1.1 Heat Equation
1.2 Fourier coefficients and Fourier series
1.3 Convergence of Fourier series
1.4 Riemann-Lebesgue theorem
1.5 Bessel inequality and Parseval identity
1.6 Fejér's Theorem
1.7 Applications: isoperimetric problem, Weyl equidistribution, heat and vibrating string equations

2. Fourier transform
2.1 Fourier transform in Schwartz space
2.2 Convolution
2.3 Inverse
2.4 Fourier transform on L^1 and L^2
2.5 Plancherel's Theorem
2.6 Applications to partial derivative equations

3. Discrete Fourier transform
3.1 Definition and properties
3.2 Convolution
3.3 Inverse
3.4 Fast Fourier Transform
3.5 Applications to Number Theory (Dirichlet's Theorem)
Main Bibliography • Iório, R & Iório, V., Equações Diferenciais Parciais: uma Introdução, Projeto Euclides, IMPA
• Churchill, R. V., & Brown, J. W. (1993). Fourier series and boundary value problems. McGraw-Hill
• Figueiredo, D. G. (2003), Análise de Fourier e Equações Diferenciais Parciais, Projeto Euclides, IMPA
• Girão, P.M. (2014), Análise Complexa, Séries de Fourier e Equações Diferenciais, IST Press
• Osgood, B. G. (2019). Lectures on the Fourier transform and its applications (Vol. 33). American Mathematical Soc..
• Stein, E. M., & Shakarchi, R. (2011). Fourier analysis: an introduction (Vol. 1). Princeton University Press.
• Vretblad, A. (2003). Fourier analysis and its applications (Vol. 223). Springer Science & Business Media.
Teaching Methodologies and Assessment Criteria The teaching of this Curricular Unit is based on the presentation by the teacher of the contents and respective bibliography, followed by the resolution of exercises and open discussion of related problems. Periodically, lists ofexercises/problems are presented to students, which are corrected and submitted for discussion. A theme is proposed to be developed by each student and presented in the classroom.

Language Portuguese. Tutorial support is available in English.
Last updated on: 2024-03-11

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