SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1374-1398, January 2021.
Based on the Hilb type formula between Jacobi polynomials and Bessel functions, optimal decay rates on the 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 parameters, the higher convergence rates can be achieved. In particular, the truncated error of Legendre projection has one half order higher than that of Chebyshev projection. 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 $\lambda$.

中文翻译：

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代数和对数正则函数的谱正交投影逼近的收敛速度

SIAM数值分析学报，第59卷，第3期，第1374-1398页，2021年1月。
根据雅可比多项式和贝塞尔函数之间的希伯型公式，通过对代数和对数奇异函数应用范德·科珀特引理，得出雅可比展开系数的最优衰减率，从而得出雅各比，格根鲍尔的最优收敛率和切比雪夫正交投影。有趣的是，对于边界奇异点，随着$（\ alpha，\ beta）$和$ \ lambda $的增加，可能会在Jacobi或Gegenbauer投影上获得更快的收敛速度。参数值越大，收敛率越高。特别地，勒让德投影的截断误差比切比雪夫投影的截断误差高一半。此外，如果$ \ min \ {\ alpha，\ beta \}> 0 $和$ \ lambda> \ frac {1} {2} $，与Legendre相比，Jacobi和Gegenbauer正交投影的收敛阶数更高。对于内部奇异点，收敛顺序与$（\ alpha，\ beta）$和$ \ lambda $无关。