Fast, numerically stable computation of oscillatory integrals with stationary points

We present a numerically stable way to compute oscillatory integrals of the form $\int{-1}^{1} f(x)e^{i\omega g(x)}dx$. For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane

Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude

The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.Comment: 36 page

Oscillatory integrals with uniformity in parameters

We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number theory.Comment: Final version. To appear in Journal de Th\'eorie des Nombres de Bordeaux. Portions of this work originally appeared in arXiv:1608.06854 (Petrow-Young) and arXiv:1701.07507 (Kiral-Young). arXiv admin note: text overlap with arXiv:1701.0750

Singular Oscillatory Integrals on R^n

Let Pd,n denote the space of all real polynomials of degree at most d on R^n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P in Pd,1. Using this estimate, we prove a sharp estimate for a singular oscillatory integral on R^n.Comment: final version, 10 pages, small typos corrected, one reference added. To appear in Math.

Uniform estimates for cubic oscillatory integrals

This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs.Comment: 22 pages; v2 added reference

Global Range Estimates for Maximal Oscillatory Integrals with Radial Testfunctions

We consider the maximal function of oscillatory integrals and prove a global estimate for radial test functions which is almost sharp with respect to the Sobolev regularity.Comment: Preprints in Mathematical Sciences 2011:1 ISSN 1403-9338, LUNFMA-5068-201