When solving differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials $T_{n}(x)$ with coefficients $a_{n}$ to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, $u(\pm 1)=0$, popular choices include the ``Chebyshev difference basis", $ς_{n}(x) \equiv T_{n+2}(x) - T_{n}(x)$ with coefficients here denoted $b_{n}$ and the ``quadratic-factor basis functions" $\varrho_{n}(x) \equiv (1-x^{2}) T_{n}(x)$ with coefficients $c_{n}$. If $u(x)$ is weakly singular at the boundaries, then $a_{n}$ will decrease proportionally to $\mathcal{O}(A(n)/n^κ)$ for some positive constant $κ$, where the $A(n)$ is a logarithm or a constant. We prove that the Chebyshev difference coefficients $b_{n}$ decrease more slowly by a factor of $1/n$ while the quadratic-factor coefficients $c_{n}$ decrease more slowly still as $\mathcal{O}(A(n)/n^{κ-2})$. The error for the unconstrained Chebyshev series, truncated at degree $n=N$, is $\mathcal{O}(|A(N)|/N^κ)$ in the interior, but is worse by one power of $N$ in narrow boundary layers near each of the endpoints. Despite having nearly identical error \emph{norms}, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic-factor and difference basis sets is nearly uniform oscillations over the entire interval in $x$. Meanwhile, for Chebyshev polynomials and the quadratic-factor basis, the value of the derivatives at the endpoints is $\mathcal{O}(N^{2})$, but only $\mathcal{O}(N)$ for the difference basis.
Hermite spectral approximation for functions with endpoint singularities using exponential transforms
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces
Department of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学系）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院数学科学分部）；State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所计算机科学与技术国家重点实验室/并行计算实验室）
