报告主题：谱逼近理论的最新进展:分数次Sobolev 空间的Jacobi逼近
报告人：Wang Li-Lian 副教授 （南洋理工大学 数学系）
报告时间：2017年5月23日（周二）9:00
报告地点：校本部G507
邀请人：马和平
主办部门：8455新葡萄场网站数学系
报告摘要：Most of spectral approximation results, if not all, are bounded by usual (weighted) Sobolev norms involving integer order derivatives. We understand the fractional Sobolev norms perhaps through space interpolation or in the sense of Gagliardo's fractional Sobolev spaces (1958). Here, we derive some fractional Sobolev spaces involving Riemann-Liouville fractional integrals/derivatives naturally from estimating Jacobi polynomial expansions. More precisely, we define the generalised Jacobi functions of fractional degree (GJF-Fs) by allowing the (integer) degree of the Jacobi polynomials (JPs) defined by the hypergeometric functions to be real. The significance of this family of GJF-Fs resides in that they enjoy some fractional integral/derivative formulas, leading to analytical tools for both algorithm development and analysis of fractional PDEs. Using the fractional integration by parts, the Jacobi polynomial expansion coefficients for functions with fractional Sobolev regularity can be precisely expressed in terms of GJF-Fs. We are then able to optimally estimate the decay rate of the expansion coefficients, so do the errors of the orthogonal projections, polynomial interpolation and quadratures.
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