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.
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.