Laurea magistrale in Matematica

The purpose of the course is to show how two fundamental tools of Mathematical Analysis and its applications, such as Fourier series and Fourier transform, find an elegant unifying conceptual framework in Abstract Harmonic Analysis. The acquired competencies relate to the learning and the use of some tipical methods for Harmonic Analysis.

INDICATORS OF DUBLIN (in reference to the Academic Regulations, European descriptors of the qualification-"Dublin descriptors"

http://www.study-in-italy.it/php5/scheda_corso.php?ambiente=offf&anno=2009&corso=1214968)

Knowledge and understanding. The course introduces the basic tools of modern harmonic analysis and Fourier analysis from a general point of view. In particular, the course is spread over three levels: Fourier analysis in R^d, harmonic analysis in L^1(G) with G a topological group, and finally an introduction to the spectral theory of commutative Banach algebras. The overall understanding of the topics requires and develops various skills specific of functional analysis, algebra and topology (targets 12, 13, 20).

Applying knowledge and understanding. The course is divided into a series of theorems with their proofs carried out in class and others left to the student. In these latter proofs the student is asked to independently produce the steps necessary to obtain rigorous mathematical results which are not identical but similar to those already known (objective 1). It is also proposed to the student to solve problems of moderate difficulty relating to examples covered in the course (objective 2).

Making judgments. The proofs of the theorems presented in the course require students to be able to build and develop logical arguments with a clear identification of assumptions, deductions and conclusions (Objective 1). It also requires students to be able to distinguish between rigorous proofs from those based on intuition, and obviously from the incorrect ones, whose errors, in this context, are often insidiously hidden in the non-applicability of right results in the specific situation (objective 2).

Communication skills. The exam preparation as well as the discussions during the course require and stimulate the development of non-trivial communication skills aimed at the precise description of abstract objects, ideas and possible solutions and/or changes in the issues addressed (Objective 1).

Learning skills. The course provides an example of "the construction of a mathematical theory" (the theory of Gelfand transform of commutative Banach algebras) that is sufficiently general to include signifcative special cases, and at the same time sufficiently rich and deep to be able to interpret the essence the objects it deals with. The awareness of these two contrasting aspects, combined with the ability to understand the technical details of the mathematical building itself, is an essential element in the development of a mature mathematical mentality.

The course leads to a thourough understanding of the tipical tools of Harmonic and Fourier Analysis. This allows the student to furhter approach some of the most modern and active reasearch areas in mathematics, namely, non-commutative harmonic analysis, operator algebras, time-frequecy analysis.

Oral text: theory and proofs presented in the corse will be required, no exercise will be required.

-       Banach Algebras, Gelfand Transform;

-       Locally Compact Groups;

-       Dual of a Locally Compact Group,

-       Fourier Transform on Locally Compact Groups.

- G. Folland, A Course in Abstract Harmonic Analysis, CRC Press 1995.

- W. Rudin, Fourier Analysis on Groups, Wiley 1990