  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.
