A general framework is introduced to analyze the approximation
properties of mapped Legendre polynomials and of interpolations based
on mapped Legendre-Gauss-Lobatto points. Optimal error estimates
featuring explicit expressions on the mapping parameters for several
popular mappings are derived.  These results not only play an
important role in numerical analysis of mapped Legendre spectral and
pseudospectral methods for differential equations, but also provide
quantitative criteria for the choice of parameters in these mappings.