Based on the Hilb type formula between Jacobi polynomials and Bessel functions, optimal decay rates on Jacobi expansion coefficients are derived, by applying van der Corput type lemmas, for functions of algebraic and logarithmatic singularities, which leads to the optimal convergence rates on the Jacobi, Gegenbauer and Chebyshev orthogonal projections. It is interesting to see that for boundary singularities, one may get faster convergence rate on the Jacobi or Gegenbauer projection as $(\alpha,\beta)$ and $\lambda$ increases. The larger values of parameter, the higher convergence rates can be achieved. In particular, the Legendre projection has one half order higher than Chebyshev. Moreover, if $\min\{\alpha,\beta\}>0$ and $\lambda>\frac{1}{2}$, the Jacobi and Gegenbauer orthogonal projections have higher convergence orders compared with Legendre. While for interior singularity, the convergence order is independent of $(\alpha,\beta)$ and $\lambda$. Furthermore, rational barycenteric interpolation based on the roots of orthogonal polynomials are introduced to fast approximation of functions and their derivatives of singularities.

向淑晃，中南大学数学与统计学院教授、博士生导师、湖南省计算数学与应用软件学会理事长，主要从事高频振荡问题、正交多项式理论等研究，2006年入选教育部新世纪优秀人才计划，2011年入选湖南省学科带头人培养对象；2003年9月-2004年9月在英国剑桥大学访问，2004年11月-2005年9月获日本JSPS资助任弘前大学长期特邀研究员，2008年9月-2009年9月香港理工大学研究员。在SIAM J. Numer. Anal.、SIAM J. Sci. Comput.、SIAM J. Optimization、Math. Program.、Numer. Math.、Math. Comput.等国际计算数学顶级期刊发表系列论文，Wang-Xiang给出的有关高斯-勒让德多项式零点与积分权的重心插值公式被国际权威Trefethen称为多项式关键十一个公式之一 ，高频振荡问题的研究也成为国际上几个重要团队之一。