报告题目：
Error analysis of Gegenbauer approximations in fractional spaces using fractional Peano kernel
报告人：刘文杰
报告时间: 2025年5月5日下午4:00
报告地点: 开元校区工科4-605
主办单位: 数学与统计学院
报告摘要：Approximation by Gegenbauer polynomials can be of great value for numerical analysis and computational algorithms, such as for boundary value problems in partial differential equations (PDEs). As far as we know, there are few rigorous proofs of Gegenbauer approximations for functions with interior or endpoint sigularities in fractional spaces. In this paper, optimal error estimates for functions of limited regularity expanded in terms of Gegenbauer polynomial series are presented based on the fractional spaces involving Riemann-Liouville (RL) fractional integrals and Caputo fractional derivatives. Such spaces are naturally arisen from exact representations of Gegenbauer expansion coefficients. We present new fractional Peano kernel formulas which enable us to obtain the optimal decay rate of Gegenbauer expansion coefficients. For a class of functions with interior or endpoint singularities, we derive the optimal (weighted) $L^\infty$-estimates and $L^2$-estimates of the Gegenbauer polynomial approximations. We provide ample numerical experiments to demonstrate the optimality and sharpness of the estimates.
报告人简介：刘文杰，哈尔滨工业大学数学学院副教授，博士生导师。2012年和2016年于哈尔滨工业大学数学学院获得硕士学位和博士学位，2016年11月至2017年11月在新加坡南洋理工大学从事博士后研究。主要研究方向为具奇异性问题的多项式逼近理论、具奇异性偏微分方程谱元法的误差估计等。发表学术论文20余篇，部分论文发表于 Mathematics of Computation、 Journal of Approximation Theory、 Journal of Computational Physics 和Journal of Scientific Computing等。其研究工作获得国家自然科学基金面上项目、天元东北中心优秀青年学者奖励计划、黑龙江省优秀青年基金项目、中国博士后科学基金面上项目（一等资助）和国家自然科学基金青年项目的资助。
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