Motivated by questions on the preconditioning of spectral methods, and
independently of the extensive literature on the approximation of
zeroes of orthogonal polynomials, either by the Sturm method, or by the
descent method, we develop a stationary phase-like technique for
calculating asymptotics of Legendre polynomials.
The difference with the classical stationary phase method is that the
phase is a nonlinear function of the large parameter and the integration
variable, instead of being a product of the large parameter by a
function of the integration variable. We then use an implicit
functions theorem for approximating the zeroes of the derivatives of
Legendre polynomials.
This result is used for proving order and consistency of the residual
smoothing scheme [1], [19].