Unit APPROXIMATION THEORY

Course
Mathematics
Study-unit Code
55A00075
Curriculum
Matematica per l'economia e la finanza
Teacher
Gianluca Vinti
Teachers
  • Gianluca Vinti
Hours
  • 63 ore - Gianluca Vinti
CFU
6
Course Regulation
Coorte 2022
Offered
2022/23
Learning activities
Caratterizzante
Area
Formazione teorica avanzata
Academic discipline
MAT/05
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
•Introduction.
• The Best Approximation Problem:
-Best approximation in normed linear spaces
Polynomial approximation
-Best uniform approximation
- Bernstein's polynomials
-First Weiestrass Theorem
-Moduli of continuity and of smoothness , estimates of error of approximation and K-Functionals
-Bohman-Korovkin's Theorem and its generalizations
- Trigonometric polynomials
- Second Weiestrass's Theorem
- Uniform approximation with trigonometric polynomials
• Brief description of spline function interpolation
• Jackson's theorems:
- Direct theorems
- Inverse teorems

• Linear integral operators:
- L ^ P spaces and dense subsets in L ^ p: short recalls
-Convolution operators and approximate identities
- Singular integrals
- Strong convergence criteria
- positive kernels
- Pointwise convergence and almost everywhere convergence
-Convergence in variation
- Order of approximation
-Lipschitz classes and saturation classes (Favard)
• Approximate sampling theorem:
-recall of the sampling theorem
-Approximate version and prediction
- Pointwise and uniform convergence
Order of Approximation
-Convergence in L ^ P and generalizations
-Sampling-Kantorovich operators: estimates and convergence
-Multi-dimensional extensions
Reference texts
- Handouts by the teacher

- P.L. Butzer-R.J. Nessel, “Fourier Analysis and Approximation, vol I”, Academic Press, 1971.

- G.G. Lorentz “Approximation of functions”, AMS Chelsea Publishing, 1986.

- W. Cheney – W. Light “A course in Approximation Theory”, Graduate Studies in Mathematics, Vol 101, Amer. Math. Soc., 2000.
Educational objectives
The course represents the first teaching of the Master Degree in Mathematics which are supplied to the main instruments that describe the processes of analytical approximation of functions.

The goal of the course is to provide students with the theory and tools for knowledge of analytical approximation processes, analyzing also applied aspects.

The main knowledge gained will be:

- Knowledge of the convolutions and their property;

- The singular integral in one and multidimensional version;

- The polynomial approximation techniques;

- The study of positive linear operators and their main results;

- Applications to the theory of signals via the sampling theorem and the use of sampling operators for the reconstruction of signals and images in approximate form.



The main skills (ie the ability to apply their knowledge) will be:

- Understand the correct definitions, properties and techniques of the mainapproximation processes;

- Assessing the effectiveness of an approximation process in terms of rate of convergence and relatively to the problem to be solved;

- To know the potential applications of the approximation processes for the creation of mathematical models and algorithms.
Prerequisites
For the purpose of the course it is necessary to understand the knowledge of the content of the courses Functional Analysis and Fourier Analysis.
Teaching methods
The course is organized in lessons on all the topics of the course
Other information
Attendance is strongly recommended for all lessons.
Learning verification modality
The evaluation mode consists in an oral test of about 30 minutes with the aim to learn the knowledge of the student of the theoretical contents of the course together with its ability to solve some application problems where approximation processes play a key role. The exam will also allow us to test the ability of expression of the student and his rigor in the formulation of the topics of the course.
Extended program
•Introduction.
• The Best Approximation Problem:
-Best approximation in normed linear spaces
Polynomial approximation
-Best uniform approximation
- Bernstein's polynomials
-First Weiestrass Theorem
-Moduli of continuity and of smoothness , estimates of error of approximation and K-Functionals
-Bohman-Korovkin's Theorem and its generalizations
- Trigonometric polynomials
- Second Weiestrass's Theorem
- Uniform approximation with trigonometric polynomials
• Brief description of spline function interpolation
• Jackson's theorems:
- Direct theorems
- Inverse teorems

• Linear integral operators:
- L ^ P spaces and dense subsets in L ^ p: short recalls
-Convolution operators and approximate identities
- Singular integrals
- Strong convergence criteria
- positive kernels
- Pointwise convergence and almost everywhere convergence
-Convergence in variation
- Order of approximation
-Lipschitz classes and saturation classes (Favard)
• Approximate sampling theorem:
-recall of the sampling theorem
-Approximate version and prediction
- Pointwise and uniform convergence
Order of Approximation
-Convergence in L ^ P and generalizations
-Sampling-Kantorovich operators: estimates and convergence
-Multi-dimensional extension
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