Luca ESPOSITO | PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

cod. 0522200010

PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

	0522200010
	DEPARTMENT OF MATHEMATICS
	EQF7
	MATHEMATICS
	2023/2024

CURRICULUM APPLICATION-ORIENTED MATHEMATICS

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2018
	FULL ACADEMIC YEAR

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
1	ISTITUZIONI DI ANALISI SUPERIORE A
	MAT/05	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
2	ISTITUZIONI DI ANALISI SUPERIORE B
	MAT/05	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

	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.

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
	PART I (48 HOURS) - 1.TOPOLOGY OF METRIC AND STANDARD SPACES (4 HOURS). BANACH SPACES AND CONTINUOUS FUNCTION SPACES [GI] (2 HOURS). ASCOLI-ARZELÀ THEOREM (2 HOURS). (TOTAL 8 HOURS) 2.LEBESGUE MEASUREMENT AND INTEGRATION THEORY. POSITIVE BOREL MEASUREMENTS (12 HOURS). MONOTONE CLASSES AND MEASUREMENT EXTENSION THEOREM (4H) [CA]. SPACES LP [RU] (6 HOURS). CONVOLUTION AND REGULARIZATION (6 HOURS). RIESZ-FRÉCHET-KOLMOGOROV THEOREM [BR] (2 HOURS). (TOTAL 30 HOURS) 3.HILBERT SPACES [RU](10 HOURS) PART II (48 HOURS) - 1. THE COMPLEX PLAN. DERIVABILITY IN THE COMPLEX SENSE (4 HOURS). INTEGRATION IN THE COMPLEX FIELD (4 HOURS). CAUCHY INTEGRAL THEOREM (2 HOURS)[CO/GR]. (TOTAL 10 HOURS) 2. CAUCHY'S INTEGRAL FORMULA AND APPLICATIONS (4 HOURS). ANALYTICAL FUNCTIONS (2 HOURS). PRINCIPLES OF IDENTITY (2 HOURS). LAURENT SERIES (4 HOURS). CLASSIFICATION OF ISOLATED SINGULARITIES (2 HOURS). RESIDUE THEORY [CO/GR (6 HOURS)]. [CO/GR]. (TOTAL 20 HOURS) 3.FOURIER SERIES. [GI]. APPLICATION TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (PDE). (10 HOURS) 4. FOURIER TRANSFORM. L1 THEORY AND INVERSION FORMULA. L2 THEORY AND PLANCHEREL THEOREM [RU]. APPLICATION TO INITIAL VALUE PROBLEMS FOR PDE. (8 HOURS)

Contents

PART I (48 HOURS) -
1.TOPOLOGY OF METRIC AND STANDARD SPACES (4 HOURS). BANACH SPACES AND CONTINUOUS FUNCTION SPACES [GI] (2 HOURS). ASCOLI-ARZELÀ THEOREM (2 HOURS). (TOTAL 8 HOURS)
2.LEBESGUE MEASUREMENT AND INTEGRATION THEORY. POSITIVE BOREL MEASUREMENTS (12 HOURS). MONOTONE CLASSES AND MEASUREMENT EXTENSION THEOREM (4H) [CA]. SPACES LP [RU] (6 HOURS). CONVOLUTION AND REGULARIZATION (6 HOURS). RIESZ-FRÉCHET-KOLMOGOROV THEOREM [BR] (2 HOURS). (TOTAL 30 HOURS)
3.HILBERT SPACES [RU](10 HOURS)

PART II (48 HOURS) -
1. THE COMPLEX PLAN. DERIVABILITY IN THE COMPLEX SENSE (4 HOURS). INTEGRATION IN THE COMPLEX FIELD (4 HOURS). CAUCHY INTEGRAL THEOREM (2 HOURS)[CO/GR]. (TOTAL 10 HOURS)
2. CAUCHY'S INTEGRAL FORMULA AND APPLICATIONS (4 HOURS). ANALYTICAL FUNCTIONS (2 HOURS). PRINCIPLES OF IDENTITY (2 HOURS). LAURENT SERIES (4 HOURS). CLASSIFICATION OF ISOLATED SINGULARITIES (2 HOURS). RESIDUE THEORY [CO/GR (6 HOURS)]. [CO/GR]. (TOTAL 20 HOURS)
3.FOURIER SERIES. [GI]. APPLICATION TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (PDE). (10 HOURS)
4. FOURIER TRANSFORM. L1 THEORY AND INVERSION FORMULA. L2 THEORY AND PLANCHEREL THEOREM [RU]. APPLICATION TO INITIAL VALUE PROBLEMS FOR PDE. (8 HOURS)

	Teaching Methods
	THE COURSE INCLUDES THEORETICAL LESSONS, DURING WHICH THE WAY IN WHICH THE ACQUIRED KNOWLEDGE CAN BE USED FOR THE SOLUTION OF PROBLEMS CONNECTED TO THE TOPICS COVERED WILL ALSO BE ILLUSTRATED. PARTICIPATION IN THE FRONTAL TEACHING IS STRONGLY RECOMMENDED.

	Verification of learning
	THE EXAM CONSISTS OF AN ORAL TEST AIMED AT ASSESSING THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED IN THE LESSONS WITH CONCEPTUAL AND TECHNICAL QUESTIONS ON THE TOPICS CONSIDERED IN THE LESSONS. DURING THE ORAL EXAMINATION THE CANDIDATE WILL ALSO BE ASKED TO CARRY OUT AN EXERCISE OF THE SAME TYPE AS THOSE CARRIED OUT IN LESSON. HONOR MAY BE GIVEN TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY THE KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT FROM THOSE PROPOSED IN LESSON.

Verification of learning

THE EXAM CONSISTS OF AN ORAL TEST AIMED AT ASSESSING THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED IN THE LESSONS WITH CONCEPTUAL AND TECHNICAL QUESTIONS ON THE TOPICS CONSIDERED IN THE LESSONS. DURING THE ORAL EXAMINATION THE CANDIDATE WILL ALSO BE ASKED TO CARRY OUT AN EXERCISE OF THE SAME TYPE AS THOSE CARRIED OUT IN LESSON.
HONOR MAY BE GIVEN TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY THE KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT FROM THOSE PROPOSED IN LESSON.

	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]

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]

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Luca ESPOSITO | PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS

CURRICULUM APPLICATION-ORIENTED MATHEMATICS

TEACHING UNITS

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Luca ESPOSITO | PRINCIPLES OF ADVANCED MATHEMATICAL ANALYSIS