A note on the numerical evaluation of integrals over strongly oscillating functions

In the present note we consider the numerical evaluation of integrals over strongly oscillating functions of the type

$\text{∫}\text{0}\text{∞}\text{f}\text{(}\text{x}\text{)?(}\text{x}\text{)}\text{dx}$

where f(x) is periodic function and where the asymptotic behaviour of ? (x) for large values of x is given by $\text{?(}\text{x}\text{) ～}\text{c}\text{x}{}^{\mathrm{\alpha }}\text{+0(}\text{1}\text{x}{}^1\text{+α}\text{)}$ with α ? 1. We show how the integral (1) can be transformed into a sum of two integrals, one of them being of the same type but with an asymptotic expansion for ? (x) given by $\text{?(}\text{x}\text{) ～}\text{c}\text{x}{}^1\text{+α}\text{+0(}\text{1}\text{x}{}^2\text{+α}\text{)}$, the other one being a proper integral.     

On numerical evaluation of integrals involving oscillatory Bessel and Hankel functions

Numerical Algorithms - A mixed-type formulation composed of Gauss-Laguerre quadrature and meshless collocation is presented for approximation of oscillatory integrals containing Hankel function of...     

A numerical evaluation of very oscillatory infinite integrals by asymptotics

Recently, Levin and Sidi have given some nonlinear methods for the accurate evaluation of slowly converging infinite integrals. But, one needs to solve some very instable linear systems, when, for example, the integrand have an increasingly rapid oscillatory behavior at infinity. In this work, we present a direct method, which avoids the instable linear systems.     

A NOTE TO OSCILLATORY SINGULAR INTEGRALS ON HERZ—TYPE SPACES

In this paper, we want to improve our previous results. We prove that some oscillatory strong singular integral operators of non-convolution type with non-polynomial phases are bounded from Herz-type Hardy spaces to Hertz spaces and from Hardy spaces associated with the Beurling algebras to the Beurling algebrasin higher dimensions.     

On the evaluation of Cauchy principal value integrals of oscillatory functions

The problem of the numerical evaluation of Cauchy principal value integrals of oscillatory functions , where −1<τ<1, has been discussed. Based on analytic continuation, if f is analytic in a sufficiently large complex region G containing [−1, 1], the integrals can be transformed into the problems of integrating two integrals on [0,+∞) with the integrand that does not oscillate, and that decays exponentially fast, which can be efficiently computed by using the Gauss-Laguerre quadrature rule. The validity of the method has been demonstrated in the provision of two numerical experiments and their results.     

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.     

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.     

Asymptotic expansions for oscillatory integrals using inverse functions

We treat finite oscillatory integrals of the form ∫ _a ^b F(x)e ^ikG(x) dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient ∫ _a ^b φ(t)e ^ikt dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when , −1<σ ₁<σ ₂<⋅⋅⋅. The authors were supported by the Office of Advanced Scientific Computing Research, Office of Science, US Department of Energy, under Contract DE-AC02-06CH11357.     

A note on finite topologies and switching functions

Finite topologies and switching functions are investigated. We associate switching functions to families of subsets of a finite set as done for instance by Adám (Truth Functions and the Problem of their Realizations by Two-terminal Graphs [Akademiai Kiadó, Budapest, 1968]); we consider then the special case where the families are (finite) topologies. We characterize switching functions which correspond to finite topologies, we associate certain functions, formed with the aid of subfunctions, to topologies on subspaces and on quotient spaces, and we use them to prove some theorems concerning these topologies and to reconstruct (in a certain weak sense) a topological space given the (quotient) spaces obtained by identifying one fixed point with each one of the others.     

A three-point formula for numerical quadrature of oscillatory integrals with variable frequency

A simple three-point formula is constructed for the evaluation of general oscillatory integrals.A rigorous derivation of the local error term is presented, and the implications to high frequency oscillations are discussed.Simple examples given include integrals with variable frequency for which the usual Filon formula would be inappropriate. For cases where Filon's formula is appropriate, the new formula appears to be computationally more efficient.The main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach. Comparison with exact results is possible from the work of Stoker [16] for a case which, whilst special in the physical sense, does not simplify the integral involved.In all cases the implementation of the formula is as straightforward as the implementation of the ordinary Simpson Rule.     

The numerical evaluation of cauchy principal value integrals with non-standard weight functions

The authors develop an algorithm for the numerical evaluation of the finite Hilbert transform, with respect to non-standard weight functions, by a product quadrature rule. In particular, this algorithm allows us to deal with the weight functions with algebraic and/or logarithmic singularities in the interval [−1, 1], by using the Chebyshev points as quadrature nodes. The practical application of the rule is shown to be straightforward and to yield satisfactory numerical results. Convergence theorems are also given, when the nodes are the zeros of certain classical Jacobi polynomials and the weight is defined as a generalized Ditzian-Totik weight. This work was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (first author) and by the Italian Research Council (second author).     

A note on an extension of extrapolative techniques for a class of infinite oscillatory integrals

Recent work on extrapolative processes for a class of infinite oscillatory integrals is extended by a procedure which involves the unknown abscissae at which the remainder integral vanishes identically. The asymptotic proximity of these abscissae to a known set is used as a basis for further extrapolation.Results show that the method can be useful for the very slowly converging integrals whose amplitudes decay only algebraically.     

A remedy for the failure of the numerical steepest descent method on a class of oscillatory integrals

In this paper we demonstrate that the numerical method of steepest descent fails when applied in a straight forward fashion to the most commonly occurring highly oscillatory integrals in scattering theory. Through a polar change of variables, however, the integral can be reformulated so that it can be solved efficiently using a combination of oscillatory integration techniques and classical quadrature. The approach is described in detail and demonstrated numerically with some oscillatory integral examples. The numerical examples demonstrate that our approach avoids the failure in some special cases, such as in an acoustic scattering model oscillatory integral with observation point located in the illuminated region. This paves the way for using the framework of numerical steepest descent methods on a wider class of problems, like the 3D high frequency scattering from convex obstacles, up to now only handled in a satisfactory way by methods due to Ganesh and Hawkins (J Comp Phys 230:104–125, 2011).     

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.     

On the numerical evaluation of singular integrals

Product-integration rules of the form _–1 ¹ k(x)f(x)dx _i ⁼¹ⁿ w _ni f(x _ni ) are studied, with the points {w _ni } chosen to be the zeros of certain orthogonal polynomials, and the weights {w _ni } chosen to make the rule exact iff is any polynomial of degree less thann. If, in particular, the points are the Chebyshev points, and ifk L _p [–1, 1] for somep>1, then it is shown that the rule converges to the exact result for all continuous functionsf. With this choice of points, the practical application of the rule is shown to be straightforward in many cases, and to yield satisfactory rates of convergence. The casek(x)=|–x|, >–1, is studied in detail. Results of a similar, but weaker, kind are also obtained for other choices of the points {x _ni }.     

Weighted norm inequalities for oscillatory integrals with finite type phases on the line

We obtain two-weighted $L^2$ norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full $L^p(\mathbb{R})\to L^q(\mathbb{R})$ mapping properties of the oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and the first author.     

Uniform approximations to Cauchy principal value integrals of oscillatory functions

This paper presents an interpolatory type integration rule for the numerical evaluation of Cauchy principal value integrals of oscillatory integrands , where -1<τ<1, for a given smooth function f(x). The proposed method is constructed by interpolating f(x) at practical Chebyshev points and subtracting out the singularity. A numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast Fourier transform. The validity of the method has been demonstrated by several numerical experiments.     

Numerical evaluation of finite Fourier integrals

A computationally efficient algorithm for evaluating Fourier integrals ∫¹_?1?(x)e^iωxdx using interpolatory quadrature formulas on any set of collocation points is presented. Examples are given to illustrate the performances of interpolatory formulas which are based on the applications of the Fejér, Clenshaw—Curtis, Basu and the Newton—Cotes points. Initially, the formulas for nonoscillatory integrals are generated and then generalizations to finite Fourier integrals are made. Extensions of this algorithm to some other weighted integrals are also considered.