谱orthogonal projection逼近代数和数学函数的收敛速度
Convergence rates of spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities
摘要：
基于雅可比多项式与Bessel函数之间的Hilb型公式，应用van der Corput型引理，导出了奇异函数Jacobi展开系数的最优衰减率，从而得到了Jacobi、Gegenbauer和Chebyshev正交投影上的最优收敛速度.有趣的是，对于边界奇点，随着$(α,β)$和$λ$的增加，可以在Jacobi或Gegenbauer投影上获得更快的收敛速度。参数值越大，可以实现越高的收敛速度。特别地，勒让德投影比切比雪夫投影高半阶。而且，当$\min\{α,β\}>0$和$λ>\frac{1}{2}$满足条件时，Jacobi和Gegenbauer正交投影的收敛阶数比Legendre投影的收敛阶数高.而对于内部奇异性，收敛阶与$(α,β)$和$λ$无关.
英文摘要：
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 $λ$.
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