Moment-free numerical integration of highly oscillatory functions

^** Email: s.olver{at}damtp.cam.ac.uk The aim of this paper is to derive new methods for numericallyapproximating the integral of a highly oscillatory function.We begin with a review of the asymptotic and Filon-type methodsdeveloped by Iserles and Nørsett. Using a method developedby Levin as a point of departure, we construct a new methodthat utilizes the same information as a Filon-type method, andobtains the same asymptotic order, while not requiring the computationof moments. We also show that a special case of this methodhas the property that the asymptotic order increases with theaddition of sample points within the interval of integration,unlike all the preceding methods whose orders depend only onthe endpoints.     

Numerical analysis of a fast integration method for highly oscillatory functions

The integration of systems containing Bessel functions is a central point in many practical problems in physics, chemistry and engineering. This paper presents a new numerical analysis for the collocation method presented by Levin for and gives more accurate error analysis about the integration of systems containing Bessel functions. The effectiveness and accuracy of the quadrature is tested for Bessel functions with large arguments. AMS subject classification (2000)  65D32, 65D30     

Efficient integration for a class of highly oscillatory integrals

This paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval. One is a Filon-type method based on the asymptotic expansion. The other is a Clenshaw-Curtis-Filon-type method which is based on a special Hermite interpolation polynomial and can be evaluated efficiently in O(N log N) operations, where N + 1 is the number of Clenshaw-Curtis points in the interval of integration. In addition, we derive the corresponding error bound in inverse powers of the frequency ω for the Clenshaw-Curtis-Filon-type method for the class of highly oscillatory integrals. The efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results.     

A method for the integration of oscillatory functions

A method for the numerical evaluation of the integrals $$I_1 (\lambda ) = \int_{ - 1}^1 {f(x)\sin (\lambda x)dx} andI_2 (\lambda ) = \int_{ - 1}^1 {f(x)\cos (\lambda x)dx} $$ is presented. The functionf(x) is approximated by a partial sum of its Legendre series.     

A numerical method for the integration of oscillatory functions

A new method for the calculation of the integrals $$I_1 (m) = \int\limits_a^b {f(x)\sin mxdx} andI_2 (m) = \int\limits_a^b {f(x)\cos mxdx}$$ is presented. The functionf(x) is approximated by a sum of Chebyshev polynomials. The Chebyshev coefficients are then used to calculate a Neumann series approximation forI ₁(m) andI ₂(m). The numerical examples demonstrate that this method is very accurate and efficient.     

Wiener-Hopf factorization for a class of oscillatory symbols

Two classes of 2×2 matrix symbols involving oscillatory functions are considered, one of which consists of triangular matrices. An equivalence theorem is obtained, concerning the solution of Riemann-Hilbert problems associated with each of them. Conditions for existence of canonical generalized factorization are established, as well as boundedness conditions for the factors. Explicit formulas are derived for the factors, showing in particular that only one of the columns needs to be calculated. The results are applied to solving a corona problem.     

A method for indefinite integration of oscillatory and singular functions

We propose a general method for computing indefinite integrals of the form

On the evaluation of highly oscillatory finite Hankel transform using special functions

In this paper, an efficient Clenshaw–Curtis–Filon–type method is presented for approximation of the highly oscillatory finite Hankel transform \({{\int }_{0}^{1}}f(x)H_{\nu }^{(1)}(\omega x)dx\), which arises in acoustic and electromagnetic scattering problems. This method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments by using Meijer G–function and Lommel function. Moreover, the method shares the property that the higher the frequency ω, the higher the precision. In particular, for each fixed ω the method is uniformly convergent as N tends to infinity, where (N+1) is the number of Clenshaw–Curtis points c_i=(1+ cos(iπ/N))/2,i=0,? ,N. Also, the corresponding error bound in inverse powers of ω for this method for the integral is presented. The efficiency and accuracy of the proposed method are illustrated by numerical examples.     

On the convergence of the method for indefinite integration of oscillatory and singular functions

We consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in Keller [8]. To this end, we have analysed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. Properties of this operator were never investigated before. The obtained results lead us to the conclusion that the studied method is always convergent. We also give a rigorous proof of the error estimates.     

On optimal integration algorithms for functions with a known number of extrema

Translated from Programmnoe Obespechenie i Modeli Issledovaniya Operatsii, pp. 177–185, 1986.     

Canonical factorization of continuous operator functions relative to the circle

Functional Analysis and Its Applications -     

On factorization of meromorphic functions

This paper improves on the results of Noda, Y., Li Baoqing and Song Quodong, and proves the following theorem: Letf(z) be a transcendental meromorphic function. Then the set {a∈C;(z−a)f(z) is not prime} is at most a countable set.     

Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integral $F(w):={\int }_{-\infty }^{\infty }{e}^{iw(t^{K+2}+e^{i\theta }{t}^p)}g(t)dt$ for large positive values of w, , K and p positive integers with $1\le p\le K$, and $g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when $w\to +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For $p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case $p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ${\mathrm{\Psi }}_K(x_1,x_2,\text{…},x_K)$ for large values of one of its variables, say $x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form $F(w)$ for appropriate values of the parameters w, θ and the function $g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large $|x_p|$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.     

Derivative and factorization of holomorphic functions

For a holomorphic function f of bounded type on a complex Banach space E, we show that its derivative df:E→E^∗ takes bounded sets into certain families of sets if and only if f may be factored in the form f=g○S, where S is in some associated operator ideal, and g is a holomorphic function of bounded type. We also prove that the multilinear and polynomial mappings factor in an analogous way if and only if they are “K-bounded.”