论文标题
代数和对数规律函数的光谱正交投影近似的收敛速率
Convergence rates of spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities
论文作者
论文摘要
基于雅各比多项式和贝塞尔函数之间的HILB类型公式，通过应用范德尔corput类型的引理，得出了雅各比扩展系数的最佳衰减率，以对对数奇点的功能，从而导致jacobi，gegenbauer and chebyshev的最佳融合率。有趣的是，对于边界奇异性，可能会以$（α，β）$和$λ$的增加而获得Jacobi或Gegenbauer投影的速度更快。参数的较大值，可以达到较高的收敛速率。特别是，Legendre投影的一半比Chebyshev高一半。此外，如果$ \ min \ {α，β\}> 0 $和$λ> \ frac {1} {2} $，则与legendre相比，Jacobi和Gegenbauer正交预测具有更高的收敛顺序。而对于室内奇异性，收敛顺序独立于$（α，β）$和$λ$。
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 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 $(α,β)$ and $λ$ 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 $\min\{α,β\}>0$ and $λ>\frac{1}{2}$, the Jacobi and Gegenbauer orthogonal projections have higher convergence orders compared with Legendre. While for interior singularity, the convergence order is independent of $(α,β)$ and $λ$.
