一类求积公式的构造方法

本文利用 Euler-Maclaurin求和公式构造了一类求积公式 ,称为修正复合梯形公式 .它和复合梯形公式的求积节点及计算量是一样的 ,但收敛阶有很大的提高 ,特别适合于计算带有各种类型小波的数值积分 .     

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.      

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

Fibonacci求积公式对具有有界混合差分的光滑函数类的误差估计

考虑对具有有界混合差分的二元光滑函数类Ｂ＾γ，ｐ，θ的求积公式，本文证明了Ｆｉｂｏｎａｃｃｉ求积公式是渐近最优的，并求出了春误差的渐近最优价。     

Quadrature Formulas Based on the Scaling Function

The scaling function corresponding to the Daubechies wavelet with two vanishing moments is used to derive new quadrature formulas. This scaling function has the smallest support among all orthonormal scaling functions with the properties M ₂ = M ₁ ² and M ₀ = 1. So, in this sense, its choice is optimal. Numerical examples are given.This work was partially supported by DFG grant GR 1777/2, by the Grant No 201/01/1200 of the CSF, by the grant MSMT 113200007 and by the grant IGS 116/5130/1 of FP TUL.     

Quadrature Formulas for Refinable Functions and Wavelets II: Error Analysis

This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines.     

AN EXTREMAL APPROACH TO BIRKHOFF QUADRATURE FORMULAS

1. Introduction and Main ResultsIn tfor paPer we shaJl use the ddstions and notations of [3l. Let E = (e'k)7t' kt. be anincidence matrir with entries consisting of zeros and ones and satisfying lEl:= Z.,* ei* = n + 1(here we allow a zero row ). Furthermore, in wha follOws we assume that(A) E satisfies the P6lya condition(B) all sequences of E in the interior rows, 0 < i < m + 1, are even.Let Sm denote the set of poiats X = (xo, z1 l "') xm, x.+1) fOr whichand Sm its clOusure. If some O…     

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

一类广义Gauss型求积公式

基于被积函数在n次第一类和第二类Chebyshev多项式的零点处的差商,该本构造了两种Gauss型求积公式. 这些求积公式包含了某些已知结果作为特例.更重要的是这些新结果与Gauss-Turan求积公式有密切的联系.     

Quadrature formulas for classes of functions with bounded mixed derivative or difference

Quadrature formulas are considered for classes of smooth functions W _p ^r , B _p ^r , θ with bounded mixed derivative or difference. For the classes of functions indicated above, the result that quadrature formulas constructed with the help of number-theoretic methods are optimal (in the sense of order) is proved, and the optimal order of the error estimates is obtained. Project supported by the National Natural Science Foundation of China and the Doctoral Program Foundation of the State Education Commission of China.     

Numerical Quadratures for Hadamard Hypersingular Integrals

In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.     

带导数的求积公式在一重积分Wiener空间下的平均误差

讨论了利用积分中值定理当积分区间趋于零时中间点的渐进位置作为相应的节点构造的带有导数的求积公式,在一重积分Wiener测度空间的平均逼近误差.     

On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle

We study the convergence of rational interpolants with prescribed poles on the unit circle to the Herglotz-Riesz transform of a complex measure supported on [–, ]. As a consequence, quadrature formulas arise which integrate exactly certain rational functions. Estimates of the rate of convergence of these quadrature formulas are also included.This research was performed as part of the European project ROLLS under contract CHRX-CT93-0416.     

三种不同意义下的最佳求积公式之间的关系

详细讨论了函数类KWr[a,b]上Sard和N iko lsk ii意义下以及基于给定信息的最佳求积公式三者之间的关系,并且提供了一种由基于给定信息的最佳求积公式得到其它两种求积公式的方法.     

On generalized averaged Gaussian formulas

We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions ( ) we give a necessary and sufficient condition on the parameters and such that the optimal averaged Gaussian quadrature formulas are internal.

     

On families of quadrature formulas based on Euler identities

A family consisting of quadrature formulas which are exact for all polynomials of order ?5 is studied. Changing the coefficients, a second family of quadrature formulas, with the degree of exactness higher than that of the formulas from the first family, is produced. These formulas contain values of the first derivative at the end points of the interval and are sometimes called “corrected”.     

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.     

Product Gaussian Quadrature on Circular Lunes

Resorting to recent results on subperiodic trigonometric quadrature, we provide three product Gaussian quadrature formulas exact on algebraic polynomials of degree $n$ on circular lunes. The first works on any lune, and has $n^2 +\mathcal{O}(n)$ cardinality. The other two have restrictions on the lune angular intervals, but their cardinality is $n^2/2 +\mathcal{O}(n)$.     

Optimal quadrature formulas for Fourier coefficients in $W_2^{(m,m-1)}$ space

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $W_2^{(m,m-1)}[0,1]$ space for calculating Fourier coefficients. Using S.~L.\ Sobolev''s method we obtain new optimal quadrature formulas of such type for $N 1\geq m$, where $N 1$ is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $m=1$. The obtained optimal quadrature formula in the $W_2^{(m,m-1)}[0,1]$ space is exact for $\exp(-x)$ and $P_{m-2}(x)$, where $P_{m-2}(x)$ is a polynomial of degree $m-2$. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring’s problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz–Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.