Course Category：Professional Basic Course
Prerequisites：advanced mathematics, linear algebra.
Aims & Requirements：
This course is one of the speciality foundation courses for master degree candidates majoring in computational mathematics. It can also be taken as an elective course for those majoring in signal processing, computer graphics and other related fields. This course principally introduces the mathematical foundations, computational methods and current trends of numerical approximation research. Participants of the course are expected to have a command of the basic theories and methods of approximation.
Chapter 1 Convolution approximation
Weierstrass approximation theorem; convolution approximation; Dirac sequence.
Chapter 2 polynomial approximation
Polynomial interpolation in one variable; polynomial interpolation in several variables.
Chapter 3 quadratic approximation
Fourier series; system of orthogonal functions; generalized Fourier series; orthogonal polynomials.
Chapter 4 Numerical quadrature
Newton-Cotes formula; Euler-Maclaurin formula; quadrature formulas of Gaussian type.
Chapter 5 Non-linear approximation
Padé approximation; rational approximation; optimal reconstruction.
Chapter 6 Spline functions
Spline functions of a single variable; B-spline functions; multi-variable splines.
Chapter 7 Wavelets
Frame theory；multi-scale analysis; construction of orthogonal wavelet basis.
WANG Renhong, Numerical Approximation, Higher Education Press,1999.
 W. Cheney and W. Light, A Course in Approximation Theory，China Machine Press,2004.
 E. M. Stein and R. Shakarchi, Fourier Analysis--An Introduction, Princeton University Press, 2003.
Author：Zhiqiang Xu (Academy of Mathematics and Systems Science)