Elements of Functional analysis and applications to Fourier series and transforms (continuous and discrete). Distributions and tempered distributions. Transformates of distributions. Elements of Lebesgue's integration theory.
Asymptotic behavior and oscillations of solutions of a differential equation. Oscillation and Comparison Theorems for Linear II order EDOs. Sturm-Liouville boundary problems.
Lebesgue's theory: Carlo Pucci - Istituzioni di analisi superiore, UMI 2013; W. Rudin - Real and complex analysis. McGraw-Hill
First Elements of Functional Analysis, Fourier Series and Transformed in L^1 and L^2:F. Mugelli - M. Spadini, Metodi Matematici. Società editrice Esculapio.
Sampling Theorem: M. Marini, Metodi Matematici per lo studio delle reti elettriche. CEDAM
Distributions and Their Transformations, Sturm-Liouville Theory and Green Function: G. C. Barozzi, Matematica per l'ingegneria dell'informazione. Zanichelli.
Learning Objectives
The training goal is the acquisition of a good disposition to the theoretical approach and to the logical-formal rigor through the elaboration of some concepts relating to Functional Analysis, Fourier theory and series transformations, distribution theory and qualitative analysis of differential equations.
Prerequisites
The contents of the courses of Analisi Matematica 1 and 2, and Geometria
Teaching Methods
Classroom-taught lessons
Further information
See
http://www.dma.unifi.it/~spadini/dida.php
Type of Assessment
oral examination
Course program
Elements of real and functional analysis
* Lebesgue measure and integral. Limit theorems.
* Normed and metric spaces. Spaces equipped with a scalar or Hermitian product. Inequality of Cauchy-Schwarz. L^p Norms. Complete metric spaces. Complete orthonormal systems and Hilbert spaces.
* The problem of the best approximation in L^2 and Fourier series.
* Fourier transform of a L^1(R)-function as a "continuous" version of the Fourier series and its properties. Inversion theorem (with proof). Duality Formula. The Fourier transform in L^2(R) (Fourier-Plancherel).
* Sampling theorem (Shannon) with proof.
* The discrete Fourier transform; interpretation as approximation; matrix form. Inversion theorem for the discrete Fourier transform.
* Space D(R) of test functions and distributions D'(R). Injectivity of the immersion of L^1_{loc} in D '(R). The Dirac delta and the principal value of 1 / x. Distribution Operations. Derived from a distribution.
* Schwartz S(R) space and tempered distributions S'(R). Inclusion of L^1(R), L^2 (R), L^infty(R) and D(R) in S. Slow growth functions. Fourier transform of a tempered distribution and its main properties.
* Concept of periodic distribution. A series of convergent distributions. Support a distribution. The notion of Laplace's transformation of a distribution. The notion of convolution of two distributions.
Boundary-value problems for ordinary differential equations: qualitative theory
Asymptotic behavior of solutions of a differential equation.
Oscillations of solutions of a differential equation.
Sturm separation theorem.
Oscillation and Comparison Theorems for Linear Differential Equations of Order II and Applications.
Common Sturm-Liouville boundary problems.
Oscillation theory for the Schroedinger equation one dimensional.