University of Chicago

Spring 2014

This is an introductory course to approximation theory, i.e., the
study of how functions can be approximated by simpler functions or linear
combinations of simpler functions. It will start with classical topics but
will gradually progress to more recent advances. The objective is to cover
a broad range of topics at the expense of giving an in-depth treatment to
only a small handful of them. We will introduce the notions of Hamel
basis, Schauder basis, orthonormal basis, dual basis, biorthogonal basis,
Riesz basis, frame, dictionary, coherence, restricted isometry property,
Haar condition, Weierstrauss, Stone–Weierstrauss,
Bohman–Korovkin and Müntz–Szász theorems, Gram
determinant, Mercer kernel, reproducing kernel Hilbert space. We will
discuss specific bases/dictionaries including: Taylor series, Fourier and
generalized Fourier series, Chebyshev and orthogonal polynomials, splines,
Gabor functions,
wavelets (also beamlets, ridgelets, curvelets, bandelets, chirplets,
noiselets). As
for algorithms, we will examine uniform and least-squares approximations,
Padé approximation, nonlinear
approximation, best *r*-term approximation, greedy approximation,
discrete
cosine, Fourier and wavelet transforms, FFT and fast wavelet transform.
Last but not least, we will look briefly at a few applications including
compression, interpolation (Vandermonde, Lagrange, Newton, Chebyshev,
Hermite, spline), quadrature, denoising, compressive sensing,
matrix
completion, implicit Runge–Kutta, finite-element and spectral
methods, support vector machines, MP3, WMA, JPEG, JPEG 2000, DjVu,
TrueType and Type1 fonts, NURBS, etc.

- 06/18/14: Scribe notes for Lecture 20 posted.

- 06/12/14: Scribe notes for Lecture 19 posted.

- 06/07/14: Scribe notes for Lecture 18 updated.

- 06/06/14: Scribe notes for Lecture 18 posted.

- 06/04/14: Scribe notes for Lecture 17 posted.

- 05/30/14: Scribe notes for Lecture 16 posted.

- 05/26/14: Scribe notes for Lecture 15 posted.

- 05/24/14: Scribe notes for Lecture 12 posted.

- 05/23/14: Make-up lecture Fri, May 30, 4:30–6, in Eckhart 133.

- 05/19/14: Scribe notes for Lectures 11 & 14 posted.

- 05/17/14: Scribe notes for Lecture 13 posted.

- 05/07/14: Guest lectures by Debashis Mondal (May 12) and Mike Stein (May 14).

- 05/04/14: Scribe notes for Lecture 9 posted.

- 05/03/14: Scribe notes for Lecture 10 posted.

- 04/25/14: Scribe notes for Lectures 6 & 8 posted.

- 04/23/14: Scribe notes for Lecture 7 posted.

- 04/21/14: Scribe notes for Lecture 5 posted.

- 04/16/14: Scribe notes for Lecture 4 posted.

- 04/09/14: Scribe notes for Lecture 3 posted. Scribe notes for Lecture 2 updated.

- 04/08/14: Scribe notes for Lecture 2 posted.

- 04/01/14: Scribe notes for Lecture 1 posted.

- 03/31/14: Check back regularly for announcements.

**Location:** Math-Stat
Building (Stevanovich Center), Room 112

**Times:** 1:30–2:50pm on Mon/Wed.

**Instructor:** Lek-Heng
Lim

Office: Eckhart
122

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

**Office hours:** By appointment.

**Grade composition:** No in-class examination. Grade based entirely
on participation in class and homework assigments.

- E.W. Cheney and W.A. Light, A Course in Approximation Theory, AMS, 2009.

- F. Cucker and D.-X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge, 2007.

- F. Deutsch, Best Approximation in Inner Product Spaces, Springer, 2001.

- S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd Ed, Academic Press, 2009.

- M.J. Mohlenkamp and M.C. Pereyra, Wavelets, Their Friends, and What They Can Do for You, EMS, 2008.

- M.J.D. Powell, Approximation Theory and Methods, Cambridge, 1981.

- V. Temlyakov, Greedy Approximation, Cambridge, 2011.

- L.N. Trefethen, Approximation Theory and Approximation Practice, SIAM, 2013.