Constructive Approximation Theory

Recommended Prerequisites

Basic courses on Functional Analysis and Complex Analysis.

Teaching Methods

The classes are essentially of expository nature and should include examples that lead the students to understanding and applying the material being taught. The classes should be focused on the teaching of the reasoning processes, so that the students learn how to manipulate the objects presented along the course and more easily find out by themselves how to reach other results by independent reading or problem solving. Some tutorial support will be available to help the students with the proposed tasks outside the classroom.

Learning Outcomes

The aim of the course is to present the approximation theory, focusing on some of their multiple subjects and recent developments. In the description of the different topics an interdisciplinary trace will be pointed out, with special mention to applications in some areas of knowledge such as approximation theory, number theory, or mathematical physics, among others. Several open problems will be described in the framework of the developed theory. The course aims at developing the following skills: knowledge of mathematical results; ability to generalize and abstract; logical thinking; competence in using computational tools. On the personal level it also allows to develop self-learning skills and independent thinking.

Work Placement(s)

No

Syllabus

The course will explore three connected areas in the framework of Approximation Theory: Orthogonal Polynomials, Special Functions, and Function Spaces. The specific topics to be studied include orthogonal polynomials, special functions (including Mellin transform), and function spaces (interpolation and embeddings).

Assessment Methods

Assessment 2
Frequency: 100.0%

Assessment 1
Exam: 100.0%

Bibliography

B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 & 2, American Mathematical Society, 2004.

E. Stein, Singular Integral and Differentiability properties of Functions, Princeton University Press, 1979.

H. Triebel, Theory of Function Spaces, Birkhauser (reprint of 1st ed.), 1983.

S. Yakubovich and Yu. Luchko, Hypergeometric Approach to Integral Transforms and Convolutions, Ser. Mathematics and its Applications, vol. 287, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.

W. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Springer-Verlag, 1989.

G.E. Andrews, R. Askey,  and R. Roy, Special Functions, Cambridge University Press, 1999.

C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988

P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, no. 3. American Mathematical Society, 2000.

R.A. DeVore and G.G. Lorentz: Constructive Approximation, Springer-Verlag, 1993.

M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.

E.M. Nikishin and V.N. Sorokin, Rational Approximations and Orthogonality, vol. 92, Translations of Mathematical Monographs, American Mathematical Society, 1991.