Asymptotic Coefficients and Errors for Chebyshev Polynomial Approximations with Weak Endpoint Singularities: Effects of Different Bases

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", $\varsigma_{n}(x) \equiv T_{n+2}(x) - T_{n}(x)$ 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 $1/n^{\kappa}$ for some positive constant $\kappa$. We prove that the Chebyshev difference coefficients $b_{n}$ decrease more slowly by a factor of $n$ while the quadratic-factor coefficients $c_{n}$ decrease more slowly still as $1/n^{\kappa-2}$. The error for the unconstrained Chebyshev series, truncated at degree $n=N$, is $O(1/N^{\kappa})$ in the interior, but is worse by one power of $N$ in narrow boundary layers near each of the endpoints. (For functions which are analytic over the whole domain including $x=\pm 1$, the Chebyshev series is much more uniform on $x \in [-1, 1]$ \cite{BoydOP63}, but the analytic-at-the-endpoints functions are not discussed here.) Although the error norms are about the same, 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 are nearly uniform oscillations over the entire interval in $x$. The value of the derivatives at the endpoints is $O(N^{2})$ for Chebyshev polynomials and for the quadratic-factor basis, but only $O(N)$ for the difference basis. In spite of these differences, there is not a compelling reason to prefer one basis over another.