Convergence Rates on Spectral Orthogonal Projection Approximation for Functions of Algebraic and Logarithmatic Regularities
Shuhuang Xiang · SIAM Journal on Numerical Analysis · 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 $\lambda$.