Quadrature formulae with free nodes for periodic functions

Summary. The problem of existence and uniqueness of a quadrature formula with maximal trignonometric degree of precision for 2-periodic functions with fixed number of free nodes of fixed different multiplicities at each node is considered. Our approach is based on some properties of the topological degree of a mapping with respect to an open bounded set and a given point. The explicit expression for the quadrature formulae with maximal trignometric degree of precision in the 2-periodic case of multiplicities is obtained. An error analysis for the quadrature with maximal trigonometric degree of precision is given. Received April 16, 1992/Revised version received June 21, 1993     

On positive positive-definite functions and Bochner’s Theorem

We recall an open problem on the error of quadrature formulas for the integration of functions from some finite dimensional spaces of trigonometric functions posed by Novak (1999) in [8] ten years ago and summarised recently in Novak and Wo?niakowski (2008) [9]. It is relatively easy to prove an error formula for the best quadrature rules with positive weights which shows intractability of the tensor product problem for such rules. In contrast to that, the conjecture that also quadrature formulas with arbitrary weights cannot decrease the error is still open.We generalise Novak’s conjecture to a statement about positive positive-definite functions and provide several equivalent reformulations, which show the connections to Bochner’s Theorem and Toeplitz matrices.     

On an optimal quadrature formula in the sense of Sard

In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K ₂(P ₂). Using S.L. Sobolev’s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L₂⁽²⁾(0,1)L_2^{(2)}(0,1). The obtained optimal quadrature formula is exact for the trigonometric functions sinx and cosx. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.     

Quadrature Formulas for Singular Integrals with Hilbert Kernel

In this paper, we first establish the quadrature formulae of proper integrals with weight by trigonometric interpolation. Then we use the method of separation of singularity to derive the quadrature formulae of corresponding singular integrals with Hilbert Kernel. The trigonometric precision, the estimate of the remainder and the convergence of each quadrature formula derived here are also established.     

On exact estimates of the convergence rate of fourier series for functions of one variable in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript>[–π, π]

The work is devoted to exact estimates of the convergence rate of Fourier series in the trigonometric system in the space of square summable 2π-periodic functions with the Euclidean norm on certain classes of functions characterized by the generalized modulus of continuity. Some N-widths of these classes are calculated, and the residual term of one quadrature formula over equally spaced nodes for a definite integral connected with the issues under consideration is found.     

带重结点的三角求积公式的迭代构造

本文以三角多项式类作为工具讨论了偶数个结点情况下的带重结点的具有最大三角精度的三角求积公式,由拟正交三角多项式的性质给出了求积公式系数的迭代构造。     

Intractability Results for Positive Quadrature Formulas and Extremal Problems for Trigonometric Polynomials

Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature formulas and state different equivalent conjectures concerning positive definiteness of certain matrices and certain extremal problems for trigonometric polynomials. We also study classes of functions with weighted norms where some variables are “more important” than others. Positive quadrature formulas are then tractable iff the sum of the weights is bounded.     

Complex measures having quadrature formulae with optimal exactness

We characterize the supports of the measures having quadrature formulae with similar exactness as Gauss’ theorem. Indeed we obtain the supports of the measures from which an m-point quadrature formula can be obtained such that it exactly integrates functions in the space ? _m?k,m?k [ $ \bar z $ , z]. We also give a method for obtaining the nodes and the quadrature coefficients in all the cases and, as a consequence, we solve the same problem in the space of trigonometric polynomials.     

Effective numerical evaluation of the double Hilbert transform

In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.     

Rational interpolation and approximate solution of integral equations

In the space of continuous periodic functions, we construct interpolation rational operators, use them to obtain quadrature formulas with positive coefficients which are exact on rational trigonometric functions of order 2n, and suggest an algorithm for an approximate solution of integral equations of the second kind. We estimate the accuracy of the approximate solution via the best trigonometric rational approximations to the kernel and the right-hand side of the integral equation.     

On general Hermite trigonometric interpolation

Summary A sequence of general Hermite trigonometric interpolation polynomials with equidistant interpolation points is given. Integrating these interpolation formulae a sequence of quadrature formulae for the integration of periodic functions is obtained. Derivative-free remainders are stated for these interpolation and quadrature formulae.This work was done at the Max-Planck-Institut für Physik und Astrophysik, München.     

Interpolation a plusieurs variables sur la sphere

Summary Given a functionf defined on the sphere , a continuation is considered which transformsf into a periodic function with period 2, accordingto each of the two usual variables , .Starting from this continuation, an explicit interpolation formula forf on in a set of trigonometric functions is obtained. A simple and numerically stable quadrature formula is given, which is accurate for a vast class of functions.Error-bounds for approximation and quadrature are given.     

On Gregory- and modified Gregory-type corrections to Newton—Cotes quadrature

Gregory-type formulae associated with the class of composite Newton—Cotes quadrature rules of the closed type are established. Furthermore, it is shown how these formulae can be extended by introducing mixed interpolation functions which contain a polynomial and a trigonometric part. The case of the modified Gregory rules associated with the composite Simpson quadrature rule is worked out in detail. Also the error term is analysed and the obtained rules are numerically tested.     

Modification of the Euler quadrature formula for functions with a boundary-layer component

The Euler quadrature formula for the numerical integration of functions with a boundary-layer component on a uniform grid is investigated. If the function under study has a rapidly growing component, the error can be significant. A uniformly accurate quadrature formula is constructed by modifying the Hermite interpolation formula so that the resulting one is exact for the boundary-layer component. An analogue of the Euler formula that is exact for the boundary-layer component is constructed. It is proved that the resulting composite quadrature formula is third-order accurate in space uniformly with respect to the boundary-layer component and its derivatives.     

Nonlinear Periodic Optimal Control: A Pseudospectral Fourier Approach

Abstract

A pseudospectral method for generating optimal trajectories of the class of periodic optimal control problems is proposed. The method consists of representing the solution of the periodic optimal control problem by an mth degree trigonometric interpolating polynomial, using Fourier nodes as grid points, and then discretizing the problem using the trapezoidal rule as the quadrature formula for smoothly differentiable periodic functions. The periodic optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the pseudospectral Fourier approach avoids many of the numerical difficulties typically encountered in solving standard periodic optimal control problems. An illustrative example is provided to demonstrate the applicability of the proposed method.     

Applications of Smolyak quadrature formulas to the numerical integration of Fourier coefficients and in function recovery problems

In this paper with the help of Smolyak quadrature formulas we calculate exact orders of errors of the numerical integration of trigonometric Fourier coefficients of functions from generalized classes of Korobov and Sobolev types. We apply the obtained results to the recovery of functions from their values at a finite number of points in terms of the K. Sherniyazov approach.     

Szegő–Lobatto quadrature rules

Gauss-type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as Gauss–Radau and Gauss–Lobatto quadrature rules, respectively. Efficient algorithms are available for their computation. Szeg? quadrature rules are analogs of Gauss quadrature rules for the integration of periodic functions; they integrate exactly trigonometric polynomials of as high degree as possible. Szeg? quadrature rules have a free parameter, which can be used to prescribe one node. This paper discusses an analog of Gauss–Lobatto rules, i.e., Szeg? quadrature rules with two prescribed nodes. We refer to these rules as Szeg?–Lobatto rules. Their properties as well as numerical methods for their computation are discussed.     

Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points.     

About some quadrature formulas

In this paper, we studied a class of quadrature formulas obtained by using the connection between the monospline functions and the quadrature formulas. For this class we obtain the optimal quadrature formula with regard to the error and we give some inequalities for the remainder term of this optimal quadrature formula.      

Prolate spheroidal wave functions on a disc—Integration and approximation of two-dimensional bandlimited functions

We consider the problem of integrating and approximating 2D bandlimited functions restricted to a disc by using 2D prolate spheroidal wave functions (PSWFs). We derive a numerical scheme for the evaluation of the 2D PSWFs on a disc, which is the basis for the numerical implementation of the presented quadrature and approximation schemes. Next, we derive a quadrature formula for bandlimited functions restricted to a disc and give a bound on the integration error. We apply this quadrature to derive an approximation scheme for such functions. We prove a bound on the approximation error and present numerical results that demonstrate the effectiveness of the quadrature and approximation schemes.