Study-unit FOURIER ANALYSIS
| Course name | Mathematics |
|---|---|
| Study-unit Code | 55A00057 |
| Curriculum | Matematica per l'economia e la finanza |
| Lecturer | Carlo Bardaro |
| Lecturers |
|
| Hours |
|
| CFU | 9 |
| Course Regulation | Coorte 2023 |
| Supplied | 2023/24 |
| Supplied other course regulation | |
| Learning activities | Caratterizzante |
| Area | Formazione teorica avanzata |
| Sector | MAT/05 |
| Type of study-unit | Obbligatorio (Required) |
| Type of learning activities | Attività formativa monodisciplinare |
| Language of instruction | Italian |
| Contents | Fourier series, Fourier transforms and application to signal processing and differential equations, theory of distributions |
| Reference texts | 1. A. Vretblad, “Fourier Analysis and its Applications”, Graduate Text in Math., 2003, Springer. 2. G.B. Folland, “Fourier Analysis and its Applications”, Pure and Applied Undergraduate text, Amer. Math. Soc., 2009. 3. L. Debnath-Bhatta “Integral Transforms and their applications”, CRC Press, Chapman & Hall, 2015 4. Didactical material of the course provided by the teacher 5. Further didactical material available in UNISTUDIUM |
| Educational objectives | Acquisition of the basic concepts of Fourier analysis, in particular of the Fourier series and the Fourier transform in L^p spaces. Primarily it is hoped that the students can begin familiar with the applications of theoretical concepts: applications for reconstruction of signals and images that have important repercussions for example in Engineering and Medicine: from the study of the structure of buildings, to the development of better medical image quality, for the accurate diagnosis of diseases. In addition, students acquire practical skills in the application of Fourier analysis to partial differential equation (the heat equation, wave equation and Laplace equation), which give models for problems in Economy and Finance. |
| Prerequisites | Elementary topics from calculus: Mathematical analysis, I and II; basic topics from measure theory and functional analysis (Hilbert and Banach spaces, main theorems) (Mathematical analysis III and IV) |
| Teaching methods | The course consists of lectures, in which in addition to the theoretical aspects, also certain practical exercises and discusses specific applications are described. It is intention of the teacher to use, if possible, even the support of some scientific seminars that highlight the practical applications. The course is developed in 63 hours of theoretical lessons, along with several applications and exercises |
| Other information | It is strongly suggested to have some familiarity with concepts from linear algebra |
| Learning verification modality | The method of evaluation includes an oral examination lasting about 30-45 minutes, in which, in addition to verifying the acquisition of the fundamental theoretical concepts, some practical exercises will be discussed, with the aim to verify the ability of the students in developing applications. The examination will also give an evaluation of the presentation skills acquired by the students. |
| Extended program | Fourier series: pointwise, uniform and L^p-convergence; finite Fourier transform and its properties. Factors of convergence and approximate identities; Fourier transform in L^1. Inversion theory, factors of convergence for the inverse transform. Approximate identities over the real line; Fourier transform in L^2 and in L^p, 1 2.; band-limited functions and Paley-Wiener Theorem; Shannon sampling theorem; Fourier transform for functions defined over an Euclidean space R^n; applications to classical differential equations: heat conduction, wave-equation, and Dirichlet problems on the half-plane; some link with other integral transforms (Laplace, Mellin, etc.). Theory of distributions: tempered distributions, Fourier transform of distributions, applications to partial differential equations. |