# Matematica | PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

## Matematica PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

 0522200010 DIPARTIMENTO DI MATEMATICA EQF7 MATHEMATICS 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 FULL ACADEMIC YEAR
 SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY 1 ISTITUZIONI DI ANALISI SUPERIORE - MODULO A MAT/05 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS 2 ISTITUZIONI DI ANALISI SUPERIORE - MODULO B MAT/05 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 LUCA ESPOSITO21
ExamDate
ISTITUZIONI DI ANALISI SUPERIORE23/06/2023 - 09:00
ISTITUZIONI DI ANALISI SUPERIORE18/07/2023 - 09:00
Objectives
THE TEACHING PROVIDES ADVANCED KNOWLEDGE AND METHODS OF MATHEMATICAL ANALYSIS OF COMMON USE IN MODERN ANALYSIS

AWARENESS: KNOW THE MEASURE THEORY AND INTEGRATION AND THE STRUCTURE OF LEBESGUE SPACES. KNOW THE THEORY OF BANACH AND HILBERT SPACES AND THE METHODS OF COMPLEX VARIABLE FUNCTIONS. KNOW THE FOURIER ANALYSIS AND THE METHODS OF THE FOURIER SERIES AND APPLICATIONS TO DIFFERENTIAL EQUATIONS. THE COURSE WILL BRING THE STUDENT TO KNOW THE SERIES IN METRIC SPACES, PROJECTIONS AND DISTANCES FUNCTION IN HILBERT SPACE, LAURENT SERIES, RESIDUES, FOURIER TRANSFORM.

COMUNICATIVE ABILITIES: THE STUDENT WILL BE ABLE TO ARTICULATE THE STATEMENTS AND THE PROOF OF THE THEOREM DEALED IN THE COURSE.

EDUCATIONAL SKILLS: THE STUDENT WILL ACQUIRE THE KNOWLEDGE OF MATEMATICAL TOOLS THAT ALLOW HIM TO HANDLE WITH MORE ADVANCED MATEMATOCAL TOPICS.
Prerequisites
KNOWLEDGE OF THE THEORY OF FUNCTIONS OF SEVERAL VARIABLES. MEASURE AND RIEMANN INTEGRALS IN R^N. BASIC TOPOLOGY.
Contents
I PART (48 H) -
1.TOPOLOGY. NORMED AND METRIC SPACES. BANACH SPACES. CONTINUOUS FUNCTION SPACES [GI]. ASCOLI-ARZEL'S THEORY.
2. MEASURE THEORY AND LEBESGUE INTEGRATION. POSITIVE BOREL MONOTONE CLASS THEOREM, EXTENSION THEOREM [CA]. L^P SPACE [RU]. CONVOLUTION AND REGULARIZATION. THE THEORY OF RIESZ-FRÉCHET-KOLMOGOROV [BR].
3.HILBERT SPEAKERS [RU]. FOURIER SERIES. [GI]. APPLICATION TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS(PDE).

II PART (48 H) -
1.COMPLEX PLANE. DERIVABILITY IN THE COMPLEX PLANE. INTEGRATION. CAUCHY INTEGRAL THEOREM [CO/GR].
2. CAUCHY INTEGRAL FORMULA AND APPLICATIONS. ANALYTIC FUNCTIONS. IDENTITY PRINCIPLES. LAURENT SERIES. CLASSIFICATION OF ISOLATED SINGULARITIES. THEORY OF RESIDUES [CO / GR]. INDEX OF A CURVA [CO]. PRINCIPLE OF THE APPLICATION. EULERO [CO / GR] FUNCTIONS.
3. FOURIER TRANSFORMATION. L1 THEORY AND INVERSION FORMULA. THEORY L2 AND PLANCHEREL'S THEORY [RU]. APPLICATION TO PROBLEMS OF INITIAL VALUES FOR PDE.
Teaching Methods
FRONTAL LESSONS
Verification of learning
THE STUDENT WILL TAKE AN ORAL EXAMINATION CONCERNING THE TOPICS COVERED IN THE COURSE.
Texts
[CA] P.CANNARSA, T.D'APRILE, INTRODUZIONE ALLA TEORIA DELLA MISURA E ALL'ANALISI FUNZIONALE, SPRINGER 2008 [CAP. 1]
[GI] E. GIUSTI, ANALISI MATEMATICA 2, BOLLATI BORINGHIERI ED. 1984 [CAP. 1; 2]
[RU] W. RUDIN, ANALISI REALE E COMPLESSA, BORINGHIERI [CAP. 1; 2; 3; 4; 9]
[BR] H. BREZIS, ANALISI FUNZIONALE (TEORIA E APPLICAZIONI), LIGUORI [CAP. 4: \$4,5]
[CO] J.B. CONWAY, FUNCTIONS OF ONE COMPLEX VARIABLE, GTM, SPRINGER-VERLAG 2ND ED. [CAP. 1; 3: \$1,2; 4; 5; 7: \$5,7,8] O IN ALTERNATIVA
[GR] D. GRECO, COMPLEMENTI DI ANALISI, LIGUORI ED. 1980 [PARTE I]