  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 $\mathcal{O}(A(n)/n^{\kappa})$ for some positive
constant $\kappa$, 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 $\mathcal{O}(A(n)/n^{\kappa-2})$. The error for the unconstrained
Chebyshev series, truncated at degree $n=N$, is
$\mathcal{O}(|A(N)|/N^{\kappa})$ 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.
