MATHEMATICAL ANALYSIS III
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- DAMIANO FOSCHI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- This is the third and last fundamental course in Mathematical Analysis for students of our three-years degree. In this course we shall present the main theorems and the principal techniques of real and complex analysis, together with some applications.
The main goal of the course is to let the student know normed spaces, L^p spaces, Hilbert spaces, Fourier series, Fourier transform, distributions, letting him/her become able to work autonomously in such spaces and with such instruments.
The second goal is to let the student understand, via examples and exercises, that these spaces and instruments are widely used not only in mathematics, but also in physics, engineering, and other sciences.
The main knowledge acquired will be:
- knowledge of L^p spaces and of the main theorems related to them
- knowledge of the concept of convolution between two measurable functions
- knowledge of the concept of linear bounded operator between Banach spaces and norm of the operator
- knowledge of the concept of Hilbert space, of the main examples of Hilbert spaces, of their properties, of the main theorems, in particular the theorem of orthogonal projections
- knowledge of the Fourier series in L^2 and its convergence properties
- knowledge of the Fourier transform in L^1 and L^2, of its properties, of the inversion theorems.
- knowledge of the concept of distribution and of the main computations with distributions
The main skills acquired will be:
- ability to identify L^p functions and to operate with it, autonomously making use of instruments like Hölder's and Minkowski's inequalities
- ability to identify a convolution and to operate with it, recognizing the support and the regularity properties of the output of a convolution
- ability to study linearity and boundedness of an operator acting between Banach spaces
- ability to recognize Hilbert spaces, to work autonomously in such spaces, in particular with orthogonal projections
- ability to compute Fourier series of piecewise regular periodic functions
- ability to compute the Fourier transform of a function and to use the transform's properties
- ability to solve autonomously partial differential equations via Fourier series or Fourier transform. Prerequisites
- Analysis I and II and basic Topology.
Course programme
- The program of the course is the following:
- Normed spaces (10 hours)
Definitions of norm and normed space. Examples of finite dimensional and infinite dimensional spaces. Norm equivalence.
Definition of Banach space.
Linear operators between normed spaces. Continuity and boundedness.
Operatorial norm. Examples of estimates fo operatorial norm.
Completeness of the space of linear bounded operators between a normed space and a Banac space.
- L^p space (10 hours)
L^p norm. Young, Hölder, Minkowski inequalities. L^\infty norm.
Definition of L^p spaces.
Inclusion properties for L^p spaces. Approximation of the L^\infty norm with L^p norms.
Completeness of L^p spaces.
Comparison of different notions of convergence: pointwise almost everywhere convergence, L^p norm convergence, uniform convergence.
Approximation properties in L^p. Density of simple functions and of continuous functions in L^p spaces.
Continuity of L^p norm with respect to translations.
Minkowski integral inequality.
Convolution product of two functions. Young's estimates for comvolution product of functions in L^p spaces.
Regularity properties of the convolution product.
Identity approximation through convolutions. Mollifiers.
Density of smooth compactly supported functions in L^p spaces.
- Hilbert spaces (10 hours)
Scalar products. Cauchy-Schwarz inequality, triangular inequality.
Parallelogram's law. Polarization formulae.
Definition and examples of Hlbert spaces.
Minimal distance point theorem.
Orthogonality. Orthogonal projections on closed subspaces.
Orthogonal decomposition of Hilbert spaces.
Riesz's representation theorem.
Orthonormal systems. Bessel's inequality.
Characterization of orthonormal bases in Hilbert spaces.
Abstract Fourier series. Parseval and Plancherel identities.
- Fourier series (8 hours)
Trigonometric polynomials.
Dirichlet kernels and Fejer kernels.
Fourier series of L^2 functions.
Completeness of the trigonometric system.
Fourier series of fondamental signals: square wave, sawtooth wave, triangular wave.
Pointwise convergence of Fourier series for piecewise regular functions.
- Fourier transform (10 hours)
How to move from Fourier series to Fourier transform.
Fourier transform for L^1 functions, definition and examples.
Riemann-Lebesgue's lemma.
Behaviour of the transform with respet to the operations of translation, dilation, derivation, multiplication by a power, convolution.
Fourier transform of gaussian functions.
Exchange of the hat formula.
Double Fourier transform. Inversion formula.
Fourier transform di Fourier for L^2 functions. Isometric properties of Fourier tansform on L^2.
Applications to the study of linear partial differential equations.
Heisenberg's indetermination principle. Didactic methods
- The course provides 48 hours of classroom lectures with presentation of theoretical aspects, applications and exercises on the blackboard.
There will also be periodic tutoring sessions (two hours per week) with exercises and review of the topics. Learning assessment procedures
- The course learning assessment is done through homework assignments and a final exam. Admission to the final exam is granted only after submission of the homework assignments.
The exam consists of two parts:
- a written examination of 3 hours, where the student is asked to solve some exercises (usually 4)
- an oral examination, consisting in a discussion about the theoretical arguments of the course.
The student is asked to discuss some of the theorems studied and of their applications.
A student is admitted to the oral examination if his mark at the written examination is greater or equal to 15. The oral examination has to be taken in the sames session as the written exam, and before the beginning of the next term's lessons.
The final mark depends on both the examinations. It isn't given by the arithmetical mean between the writtend and the oral mark, but it comes from an overall evalutation of the student's skill.
In the winter session there are at least 2 possibilities to take both the written and the oral exam, the dates are arranged with the students of the class. In the other sessions there are one or more opportunities to take the exam, depending on the studen's demand. Reference texts
- - lecture notes, with exercises, provided by the teacher
- G.Gilardi, Analisi 3, McGraw Hill
- E. Lieb & M. Loss, Analysis, American Mathematical Society
- R. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific
- W. Rudin, Real and complex analysis, McGraw-Hill
- F.Treves, Topological vector spaces, Accademic Press
- L.Zanghirati, Appunti di Analisi V, in rete
- R.Agliardi, M.Cicognani, A.Corli: Esercizi di Istituzioni di Analisi Superiore, in our library.
