Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ where the bar indicates the Cauchy principal value and $f$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $x=0$, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $\omega$ are derived for each fixed $x\geq 0$, which clarify the large $\omega$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $x$, we classify our discussion into three regimes, namely, $x=\mathcal{O}(1)$ or $x\gg1$, $0<x\ll 1$ and $x=0$. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $\omega$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.Comment: 32 pages, 6 figures, 4 table

Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis

This paper deals with a quadrature rule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis. The rule is of product type and consists in approximating the density function f by a truncated interpolation process based on the zeros of generalized Laguerre polynomials and an additional point. We prove the stability and the convergence of the rule, giving error estimates for functions belonging to weighted Sobolev spaces equipped with uniform norm. We also show how the proposed rule can be used for the numerical solution of hypersingular integral equations. Numerical tests which confirm the theoretical estimates and comparisons with other existing quadrature rules are presented

On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data

We study the Cauchy initial-value problem for the Benjamin-Ono equation in the zero-disperion limit, and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by Lax and Levermore to analyze the corresponding limit for the Korteweg-de Vries equation.Comment: 54 pages, 11 figure

Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval

We consider polynomials p^w_n(x) that are orthogonal with respect to the oscillatory weight w(x)=exp(iwx) on [?1,1], where w>0 is a real parameter. A first analysis of p^?_n(x) for large values of w was carried out in connection with complex Gaussian quadrature rules with uniform good properties in w. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of p^?_n(x) in the complex plane as n tends to infinity. The parameter w grows with n linearly. The tools used are logarithmic potential theory and the S-property, together with the Riemann--Hilbert formulation and the Deift-Zhou steepest descent method

Constructing smooth potentials of mean force, radial, distribution functions and probability densities from sampled data

In this paper a method of obtaining smooth analytical estimates of probability densities, radial distribution functions and potentials of mean force from sampled data in a statistically controlled fashion is presented. The approach is general and can be applied to any density of a single random variable. The method outlined here avoids the use of histograms, which require the specification of a physical parameter (bin size) and tend to give noisy results. The technique is an extension of the Berg-Harris method [B.A. Berg and R.C. Harris, Comp. Phys. Comm. 179, 443 (2008)], which is typically inaccurate for radial distribution functions and potentials of mean force due to a non-uniform Jacobian factor. In addition, the standard method often requires a large number of Fourier modes to represent radial distribution functions, which tends to lead to oscillatory fits. It is shown that the issues of poor sampling due to a Jacobian factor can be resolved using a biased resampling scheme, while the requirement of a large number of Fourier modes is mitigated through an automated piecewise construction approach. The method is demonstrated by analyzing the radial distribution functions in an energy-discretized water model. In addition, the fitting procedure is illustrated on three more applications for which the original Berg-Harris method is not suitable, namely, a random variable with a discontinuous probability density, a density with long tails, and the distribution of the first arrival times of a diffusing particle to a sphere, which has both long tails and short-time structure. In all cases, the resampled, piecewise analytical fit outperforms the histogram and the original Berg-Harris method.Comment: 14 pages, 15 figures. To appear in J. Chem. Phy

Dispersive and Strichartz estimates for hyperbolic equations with constant coefficients

Dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients are considered. The global time decay estimates of $L^p-L^q$ norms of propagators are obtained, and it is shown how the time decay rates depend on the geometry of the problem. The frequency space is separated in several zones each giving a certain decay rate. Geometric conditions on characteristics responsible for the particular decay are identified and investigated. Thus, a comprehensive analysis is carried out for strictly hyperbolic equations of high orders with lower order terms of a general form. Results are applied to establish time decay estimates for the Fokker-Planck equation and for semilinear hyperbolic equations.Comment: 119 page

On nonexistence of Baras--Goldstein type for higher-order parabolic equations with singular potentials

An analogy of nonexistence result by Baras and Goldstein (1984), for the heat equation with inverse singular potential, is proved for 2mth-order linear parabolic equations with Hardy-supercritical singular potentials. Extensions to other linear and nonlinear singular PDEs are discussed.Comment: 22 page