Instructor: Wenjing Liao

Office: Skiles 258

Office Hours: W 10-11AM or by appointment on Microsoft Teams, Some additional office hours are TBA.

Email: wliao60@gatech.edu

Lectures: M W 12:30-1:45PM on Microsoft Teams

Course webpage: https://people.math.gatech.edu/~wliao60/21Spring_Math6645.html

- Prerequisites: Math 4317 and Math 4640 (or equivalent)
- Course description: theoretical and computational aspects of polynomial, spline, wavelet and frame approximation, and regression in statistical learning

- Numerical Analysis Mathematics of Scientific Computing (3rd) by Kincaid and Cheney, Brooks/Cole.
- Mathematics of Approximation by Johan De Villiers, Springer Link (GT Library).
- Interpolation and approximation by polynomials, by G.M. Phillips, (GT Library, on-line).
- Numerical solution of partial differential equations by the finite element method, by Claes Johnson, Dover
- The Mathematical Theory of Finite Element Methods, by Susanne C. Brenner, L. Ridgway Scott, Springer (GT Library, on-line)
- A Basis Theory Primer by Christopher Heil, Birkhauser Link. (GT Library, on-line).
- Applied Mathematics Data Compression, Spectral Methods, Fourier Analysis, Wavelets, and Applications by Chui and Jiang, (GT Library, on-line).
- Ten Lectures on Wavelets, by Ingrid Daubechies, SIAM Link
- Introduction to Nonparametric Estimation by Alexandre B. Tsybakov, (GT Library, on-line)
- A Distribution-Free Theory of Nonparametric Regression by Laszlo Gyorfi, Michael Kohler, Adam Krzyzak and Harro Walk, (GT Library)
- "Universal algorithms for learning theory part i: piecewise constant functions", by Binev, Cohen, Dahmen, DeVore and Temlyakov, Journal of Machine Learning Research, 2005.
- "Universal algorithms for learning theory. part ii: Piecewise polynomial functions", by Binev, Cohen, Dahmen and DeVore, Constructive approximation, 2007.
- "Nonlinear approximation", by Devore, Acta Numerica, 1998.

- Homework (50%): Students are strongly encouraged to solve all the written homework problems. Exams will be based on these homework problems. Only part of the homework will be turned in. Homework are due tentatively on February 10, March 24 and April 21.
- Exams (50%=25%+25%): There will be two exams
(a midterm and a Final) during the semester which will be
based on the homework/lecture materials.

- A midterm is tentatively on March 3rd. The
midterm will be take-home, which lasts for 24 hours.

- The Final is in the exam week (April 29-May
6, specific time and location is TBA).

- No make up exams are allowed. In case of serious illness, a doctor's note is required. For excused absences, your adviser's notice is required.
- Presentation: If a student gives a
presentation on the material related with the topics of this
course, he/she is exempted from Final. The presentation will
occur at the end of the semester (around April 7-26). The
professor will select some papers in February.
Students can present either one of those papers or something
else related to his or her research. This decision should be
made and consulted with the professor before
**March 1**, and the student will give a mini 5 minute presentation around March 15-31 (the date is to be determined).

HONOR CODE: All students are expected to comply with the

- Polynomial approximations: Classical
polynomial interpolation, Chebyshev, orthogonal polynomial,
Spline, Least squares approximation --
Kincaid and Cheney Section 6.1 - 6.10

- Trigonometric polynomial approximation:
Fourier series and the FFT, transformation and their
application -- Heil Chapter 8,9,10

- Wavelet and frame approximation -- Heil Chapter 8,9,12
- Regression in statistical learning: Curse of
dimensionality, kernel methods, polynomial partitioning
estimates, adaptive methods -- Gyorfi, Kohler, Krzyzak and Walk, Chapter
1,2 and Tsybakov, Chapter 1

- Week 1, January 20

- Kincaid and Cheney, Section 6.1: Polynomial
approximation, Lagrange form, Newton form, Error analysis,
Chebyshev Polynomials and Chebyshev nodes

- Exercises
- Kincaid and Cheney, Section 6.1: 2-6,9,11,13,14,17,19,20,22,24,27,30