Asymptotic expansion of a class of one-dimensional Fourier integrals

Abstract

The method of stationary phase is used to develop an asymptotic expansion of a class of one-dimensional Fourier integrals whose phase function is a general second-degree polynomial. Because of its form, the phase function contains one stationary point that may occur anywhere on the real line. The asymptotic expansion is obtained using the theory of distributions and Fourier transform analysis. In addition to the expansion, a bound on the remainder term is derived. Finally, it is shown that the resulting expansion is, indeed, asymptotic.

Keywords:

• Asymptotic expansions
• Fourier integrals
• Fourier transforms
• Method of Stationary Phase
• Distributions

Notes

^†For a complete discussion on the method of stationary phase, the reader is referred to 2.

^†It can be shown, by induction, that C and C _n can always be found to exist.

^†In 6, the phase function was the monomial x ², hence those results were obtained by taking advantage of a situation unique to that problem—namely that is essentially its own transform. Since the phase function is a polynomial in this case, it is necessary to use the method of convolution to obtain the Fourier transform of .

^†As previously stated, in applications, one is often interested in equation (1) for large values of ω, namely ω→∞, therefore, it is reasonable to consider ω≥1/4a (a≠0).

^†The method used here to derive a bound on the remainder is the same as that used in 4. In that work results were obtained for the case, where the phase function was the monomial x ^α (α≥2)—a situation similar to the one discussed in 6.

Additional information

Notes on contributors

Paul S. Rossi

Email: [email protected]