Damping oscillatory integrals with polynomial phase.

Asymptotic expansions of oscillatory integrals with complex phase

We consider saddle point integrals in d variables whose phase functions are neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The proofs are via contour shifting in comp...

Experimental computation with oscillatory integrals

Abstract A previous study by one of the present authors, together with D. Borwein and I. Leonard [8], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high preci...

Oscillatory Gap Damping

Homogeneous Estimates for Oscillatory Integrals

Abstract. Let u(x, t) be the solution to the free time-dependent Schrödinger equation at the point (x, t) in space-time Rn+1 with initial data f . We characterize the size of u in terms Lp(Lq)-estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces Ḣs (Rn). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interp...

Shifted GMRES for oscillatory integrals

None of the existing methods for computing the oscillatory integral ∫ b a f(x)e iωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding deformation into the complex plane and insensitivity to oscillations in f . We present a new method that satisfies these properties, based on applying the gmres algorithm to a shifted linear differential operator.