Constructive Approximation Theory, January-April 2017
When Tuesday and Thursdays morning from 8:30 AM to 10:00 AM
Where Department of Computational & Data Sciences, SERC building, Room 202, Seminar Hall
Office: Department of Computational & Data Sciences, SERC building, Room 205.
Office hours: Tuesday 3:00 PM-5:00 PM or by appointment.
The course will focus on constructive approximation of real valued functions by simpler function and numerical quadrature. We will discuss the mathematical theory and computational techniques of such approximations. Error bounds and convergence of approximations will be discussed. A substantial part of the course will be dedicated to polynomial approximations and quadratures. All the course material (error bounds, convergence, Weierstrass approximation theorem, optimal interpolant, Ramez algorithm) will be discussed and illustrated computationally. The emphasis of the course is on computational assignments, which will be due weekly. The highlight of the course would be how approximations lend themselves in constructing highly accurate fast algorithms. Topics: Approximation by Algebraic Polynomials, Weierstrass Theorem, Muentz-Szasz theorem, Orthogonal polynomials, Optimal interpolation nodes, Optimal interpolant, Lebesgue constants, Fourier series, Gibbs phenomenon, Potential theory and approximation, Spectral methods, Pade approximation, Clenshaw-Curtis and Gaussian quadrature, fast low rank construction of kernel matrices and other fast matrix computations.
- Approximation Theory and Approximation Practice, by Lloyd N. Trefethen; The book is available here.
- Interpolation and Approximation by Polynomials, by George M. Phillips
- Chebyshev and Fourier spectral methods, by John P. Boyd
There will be a total of 12 homework (each worth 5 points) due roughly weekly. Homework will be posted on this website and will be due on Wednesdays before 5 PM. Late homework will not be accepted. The homework will involve fair amount of programming exercises. Students are strongly encouraged to typeset their solutions using LaTeX/TeX (10% bonus points for submitting in LaTeX/TeX). To save trees, the students need to send their homework through email to with the subject reading CAT_2017_HW_#_firstname, where # needs to be replaced with the homework number (between 1 and 12) and firstname is to be replaced with your first name in lower case. Details on how to submit the computing part of the homework will be elaborated in the homework itself.
No collaboration is allowed for homework.
The grader will expect you to express your ideas clearly, legibly, and completely, often requiring complete English sentences rather than merely just a long string of equations or unconnected mathematical expressions. This means you could lose points for poorly written proofs or answers. Clear exposition is a crucial ingredient of mathematical communication. Clarity of thought and presentation is more important in mathematics & sciences than any other field. The only way to master exposition is by repeated practicing.
If you don't have \(\LaTeX\) on your system, try any of the following online ones.
Each homework will have couple of computational exercises. Students must be comfortable with programming and are expected to have working knowledge in C++ & MATLAB. If not, they should be able to learn and immediately pick it up.
For C++ compilers, try any of the following (there any many other options if you google for "online compiler"). Pick your favorite one.
There will be a short (3-5 minutes) quiz in almost every class.
This could be a work arising out a published article or from material/discussions in class. The project is due on April 12th. Below are some articles for suggestion.
Below is a calendar, which will be updated as we make progress in the course.