The stationary Phase Method with Remainder Estimate as Dimension of the Space goes to Infinity

Abstract

Stationary phase method concerns with oscillatory integrals over R ^d of the following form: $I\left( {S,a,\nu } \right) = \int_{{{{R}^{d}}}} {{{e}^{{ - i\upsilon S(x)}}}a(x)dx} ,$ where the phase function S(x) is a real valued C ^∞ function, the amplitude α(x) is of class C ^∈, too and v is a large positive parameter. It is a method to evaluate I(S,a,v) asymptotically, as v → ∞. In the simplest case that a(x) ∈ C ^∞₀ (R ^d) and that S(x) has only one critical point x*, where HessS(x*) is non-degenerate, it gives

$$ I(S,a,v) = {{(\frac{{2\pi }}{{iv}})}^{{d/2}}}{{[\det \{ HessS(x*)\} ]}^{{ - 1/2}}}({{e}^{{ - ivS(x*)}}}a(x*) + {{r}_{d}}(v)) $$

and an estimate of the remainder term

$$ {{r}_{d}}(v) = O({{v}^{{ - 1}}}) $$

.