Description
Objectives
To master concepts and techniques in the theory of Hilbert spaces, Lp spaces, Fourier transform, distributions.
Syllabus
Review of metric spaces. Normed spaces. Hilbert spaces; orthonormal sets, hilbert bases; linear functions. Review of L^p spaces. Convergence in L^p and in measure: Egorov and Lusin theorems. Convolution in L^p; identity approximation. Duality, weak convergence; Riesz representation theorem. Radon measures. Fourier series: trigonometric series; Parseval identity; continuous functions; Fourier coefficients of L^1-functions; Riemann-Lebesgue lemma convergence theoresm, Riemann’s localisation principle, Dini’s test, Cesaro means, Fejer’s theorem. Fourier transform in Schwartz space and in Lp; convolution; diferentiation; inversion formula, Plancherel’s theorem; Hausdorff-Young inequality. Distributions, compact support and tempered; extension of the Fourier transform. Topics: Sobolve spaces; Rearrangements and integral inequalities; Applications to probability theory; Topological groups.
Prerequisites
Basic knowledge of Topology and of Measure and Integration.
Cross Competence Component
The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.
Ethical Principles
All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.