  We compare the convergence behavior of best polynomial approximations and
Legendre and Chebyshev projections and derive optimal rates of convergence of
Legendre projections for analytic and differentiable functions in the maximum
norm. For analytic functions, we show that the best polynomial approximation of
degree $n$ is better than the Legendre projection of the same degree by a
factor of $n^{1/2}$. For differentiable functions such as piecewise analytic
functions and functions of fractional smoothness, however, we show that the
best approximation is better than the Legendre projection by only some constant
factors. Our results provide some new insights into the approximation power of
Legendre projections.
