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When solving differential equations by a spectral method, it is oftenconvenient 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 choicesinclude the ``Chebyshev difference basis", $\varsigma_{n}(x) \equiv T_{n+2}(x)- T_{n}(x)$ with coefficients here denoted $b_{n}$ and the ``quadratic-factorbasis 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 $a_{n}$ willdecrease proportionally to $\mathcal{O}(A(n)/n^{\kappa})$ for some positiveconstant $\kappa$, where the $A(n)$ is a logarithm or a constant. We prove thatthe Chebyshev difference coefficients $b_{n}$ decrease more slowly by a factorof $1/n$ while the quadratic-factor coefficients $c_{n}$ decrease more slowlystill as $\mathcal{O}(A(n)/n^{\kappa-2})$. The error for the unconstrainedChebyshev 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 nearlyidentical error \emph{norms}, the error in the Chebyshev basis is concentratedin boundary layers near both endpoints, whereas the error in thequadratic-factor and difference basis sets is nearly uniform oscillations overthe entire interval in $x$. Meanwhile, for Chebyshev polynomials and thequadratic-factor basis, the value of the derivatives at the endpoints is$\mathcal{O}(N^{2})$, but only $\mathcal{O}(N)$ for the difference basis.

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