On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators

In this paper we set out to understand Filon-type quadratureof highly-oscillating integrals of the form ₀¹ f(x) e^ig(x) dx,where g is a real-valued function and >> 1. Employingad hoc analysis, as well as perturbation theory, we demonstratethat for most functions g of interest the moments behave asymptoticallyaccording to a specific model that allows for an optimal choiceof quadrature nodes. Filon-type methods that employ such quadraturenodes exhibit significantly faster decay of the error for highfrequencies . Perhaps counterintuitively, as long as optimalquadrature nodes are used, rapid oscillation leads to significantlymore precise and more affordable quadrature.     

On the Chebyshev quadrature rule of finite part integrals

The stability and the convergence of the Chebyshev quadrature rule of one-sided finite part integrals (J. Approx. Theory 111 (2001) 196–219) is considered with some numerical examples. The three-term recurrence relation is suggested for computing the quadrature weights.     

On Fourier transforms of distribution semigroups

A one-to-one correspondence is established between Fourier transforms of ultradistribution semigroups in the sense of Beurling and some class of pseudoresolvents characterized by conditions concerning their domains of existence and growth.     

On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

We deal with several classes of integral transformations of the form $$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$ , where D is an operator. In case D is the identity operator, we obtain several operator properties on L _p (?₊) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L ₂(?₊) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type.     

Modified gauss-jacobi quadrature formulas for the numerical evaluation of cauchy type singular integrals

We obtain modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy principal values of integrals α,β>?1, wheref(x) possesses one or more simple poles in (?1, 1). Forα=β=±1/2, the modified quadrature formulas are given explicitly in closed form. Examples are given to illustrate the results.     

On the numerical evaluation of Cauchy transforms

In this paper we consider simple methods for the reconstruction of the Cauchy transform over a curve when an explicit parametrization of the latter is not provided. The methods consist of replacing the parametrization of the curve by piecewise polynomial interpolation followed by the use of Newton-Cotes type formulae for the integration. The order of convergence of the resulting quadrature is higher than would be expected on the basis of considerations involving just interpolation theory, provided that the Cauchy transform is evaluated at known nodes on the curve. These results allow the calculation of the Cauchy transform at other points with the same accuracy if this scheme is followed by an interpolatory formula of sufficiently high accuracy.     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

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.     

On the representation of functions as Fourier integrals

Sufficient conditions for the representation of functions as the Fourier integral in ? ^d of a function belonging to the space L ₁ ∩ L _p , where 0 < p < 2 are obtained. The sharpness of these conditions is shown.     

Similar basis function algorithm for numerical estimation of Fourier integrals

The methodological difficulties of estimating Fourier integrals using the fast Fourier transform (FFT) algorithm have intensified the interest in an alternative approach based on the Filon’s method of computing the trigonometric integrals. Following this approach, we introduce in this paper a similar basis function (SBF) algorithm that decomposes the function to be transformed into the sum of finite elements termed “similar basis functions”. Due to a simple analytical form of SBF, the reassignment of the SBFs’ similarity relationships into the transformation domain reduces the estimation of the Fourier integrals to a number of standard computational procedures. The SBF algorithm is capable to deal with both uniform and non-uniform samples of the function under analysis. Using this opportunity, we extend a general SBF algorithm by a fast SBF algorithm which deals with exponentially increasing sampling intervals. The efficiency and the accuracy of the method are illustrated by computer experiments with frequency characteristics and transient responses of a typical dynamic system.     

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 }.     

Fast wavelet transforms and numerical algorithms I

A class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N × N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.     

一种非规则积分区域的二重高振荡函数积分方法

在传统L ev in方法与新F ilon型方法的基础上,本文提出了一种求解非规则区域下的二重高振荡函数数值积分方法,通过利用L ev in匹配法将二重积分化为一重积分,并避免了对复杂的m om en ts的求解,能提高计算的效率,且有很高的求积精度.     

On localization properties of Fourier transforms of hyperfunctions

In [A.G. Smirnov, Fourier transformation of Sato's hyperfunctions, Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space U(Rk) which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on Rk. It was shown that all Gelfand-Shilov spaces (α>1) of analytic functionals are canonically embedded in U(Rk). While the usual definition of support of a generalized function is inapplicable to elements of and U(Rk), their localization properties can be consistently described using the concept of carrier cone introduced by Soloviev [M.A. Soloviev, Towards a generalized distribution formalism for gauge quantum fields, Lett. Math. Phys. 33 (1995) 49-59; M.A. Soloviev, An extension of distribution theory and of the Paley-Wiener-Schwartz theorem related to quantum gauge theory, Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of and U(Rk) is studied. It is proved that an analytic functional is carried by a cone K⊂Rk if and only if its canonical image in U(Rk) is carried by K.