A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.     

变阶可解族的带权逼近与插值逼近

本文讨论了变阶可解逼近族的插值逼近和带权逼近(权在插值点集Z上趋于无穷而在Z外为1)的关系.指出对变阶可解族而言,当逼近解为非亏损时,稠密性假设是自然满足的,且此时的最佳插值逼近等于该带权最佳逼近的极限.     

Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich–Dickinson–Velani [6] and Vaughan–Velani [22]. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich–Zorin [5] in the divergence case.     

Spectral approximation in L(H)

We introduce the concept of numerical range approximant which is related to that of a spectral approximant. We obtain a variant of Bouldin result on spectral approximation of normal operators [3] and an analogous result for numerical range approximation . We also extend Maher's result on partially isometric approximation [9] and Halmos' on positive approximation [6].     

带约束导数值域的L逼近

1972年J.A.Roulier和G.D.Taylor研究了带约束导数值域的一致逼近,在文章最后,他们提出了一个未解决的问题,就是关于带约束导数值域的L逼近问题.本文研究了这个问题,得到与[1]平行的结果.这个结果同时也推广了 R.A.Lorentz的工作. 第一节给出存在定理,第二节证明若干特征定理,第三节给出一个唯一性定理.     

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’s original result holds for a large class of compact convex sets.     

Approximation theorems for functions of two variables in L* ϕ spaces

This paper contains three theorems on the order of approximation of 2π-periodic functions of two variables by some linear means of the Fourier series in the Orlicz spaces. These theorems generalize some results given in [1] and [5].     

Approximation processes for fourier-jacobi expansions †

This paper deals with the approximation theoretic aspects of summation methods for expansions in terms of Jacobi polynomials. When a funcation f is expanded in a Fourier-Jacobi series, many summation methods for this series may be looked upon as approximation processes for the function f. The main object of this paper is to investigate the order of approximation of these processes and to characterize the functions which allow a certain order of approximation. Many of these processes exhibit the phenomenon of saturation, which is equivalent to the existence of an optimal order of approximation (the saturation, which is equivalent to the existence of an optimal order of approximation (the saturation order). For the approximation processes treated in this paper the saturation order and the saturation class, that is the class if functions which can be approximated with the optimal order, are derived. The characterization of the classes of functions is accomplished by means of the theory of intermediate spaces due to Peetre[19] (compare Butzer and Berens [7]). Another basic tool in this work is the convolution structure for Jacobi series, introduced by Askey and Wainger [1] (see also Gasper [14], {15})     

COMPUTATION OF THE RELAXATION OSCILLATION PERIOD IN LOTKA‐VOLTERRA SYSTEMS

ABSTRACT. This work surveys techniques of Grasman and Veling [1973], Vasil'eva and Belyanin [1988] and Shih [1996] for computing the relaxation oscillation period of singularly perturbed Lotka‐Volterra systems. Grasman and Veling [1973] used an implicit function theorem to derive an asymptotic formula for the period; Vasil'eva and Belyanin [1988] employed a method of matched asymptotic expansions to obtain an approximation to the period; Shih [1996] obtained two (exact) integral representations for the period in terms of two inverse functions W(–k, x) of xexp(x). These results are compared numerically and asymptotically. In particular, the integral representation of the period in Shih [1996] is computed numerically using a Gauss‐Tschebyscheff integration rule of the first kind, and is further investigated asymptotically by virtue of the asymptotics of W(–k, x), Laplace's method, and a method of consequent representation. Computational results indicate that the Gauss‐Tschebyscheff approximation of the period in Shih [1996] is uniformly accurate for a wide range of the singular parameter (? in the paper).     

Efficient reconstruction of functions on the sphere from scattered data

Motivated by the fact that most data collected over the surface of the earth is available at scattered nodes only, the least squares approximation and interpolation of such data has attracted much attention, see e.g. [1, 2, 5]. The most prominent approaches rely on so-called zonal basis function methods [16] or on finite expansions into spherical harmonics [12, 14]. We focus on the latter, i.e., the use of spherical polynomials since these allow for the application of the fast spherical Fourier transform, see for example [8, 9] and the references therein. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.     

The projected newton method for solving inverse eigenvalue problems - the case of multiple eigenvaues

This paper deals with the optimal solution of ill-posed linear problems, i.e..linear problems for which the solution operator is unbounded. We consider worst-case ar,and averagecase settings. Our main result is that algorithms having finite error (for a given setting) exist if and only if the solution operator is bounded (in that setting). In the worst-case setting, this means that there is no algorithm for solving ill-posed problems having finite error. In the average-case setting, this means that algorithms having finite error exist if and only lf the solution operator is bounded on the average. If the solution operator is bounded on the average, we find average-case optimal information of cardinality n and optimal algorithms using this information, and show that the average error of these algorithms tends to zero as n→∞. These results are then used to determine the [euro]-complexity, i.e., the minimal costof finding an [euro]-accurate approximation. In the worst-case setting, the [euro]comp1exity of an illposed problem is infinite for all [euro]>0; that is, we cannot find an approximation having finite error and finite cost. In the average-case setting, the [euro]-complexity of an ill-posed problem is infinite for all [euro]>0 iff the solution operator is not bounded on the average, moreover, if the the solutionoperator is bounded on the average, then the [euro]-complexity is finite for all [euro]>0.     

On the approximation of the solutions of the Riemann problem for a discontinuous conservation law

For a class of discontinuous flux functions introduced in [3] (cf. also [4]), we prove, for the Riemann problem, an extension of the existence result proved in [2] for a Lipschitz continuous flux function. In the last section, and based in the previous results, we apply the Lax-Friedrichs approximation method and the limiters technique (cf.[6]) to compute the quoted solution in a numerical example. For related results see [5].     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

Approximate <Emphasis Type="Italic">k</Emphasis>-MSTs and <Emphasis Type="Italic">k</Emphasis>-Steiner trees via the primal-dual method and Lagrangean relaxation

Garg [10] gives two approximation algorithms for the minimum-cost tree spanning k vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and k-median problems. In this paper we show how Gargs algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.     

An improved implementation of the McCracken and Yanosik nine point finite difference procedure

The grid orientation phenomenon present in numerical models of fluid flow in a porous media can give rise to unrealistic predictions when simulating adverse mobility displacements. McCracken and Yanosik [11] proposed a nine point finite difference scheme for approximating the solution of the continuity equations that has the potential of eliminating many of the unrealistic predictions that are observed when using five point finite difference operators. Coats and Ramesh [5] have implemented this scheme in a steamflooding model, and have noted that serious grid effects are present in the simulation of an inverted seven spot pattern. Potempa [12,13] has described a procedure which exhibits minimal grid effects for the problem described by Coats and Ramesh [5]. This paper describes modifications to the McCracken and Yanosik procedure which allow for realistic simulations of this inverted seven spot pattern under a steam drive. These modifications are based upon an approximation scheme that has been previously reported [12,13], and affect the incorporation of upstream weighting in a similator.     

Eine Bemerkung zum Satz von Müntz

Luxemburg and Korevaar obtain in [3] a relationship between entire functions and Müntz's approximation theorem. In this note we want to show that a slightly modified version of [2] implies the main results of [3].     

Approximation in eigenvalue problems for holomorphic fredholm operator functions I

The approximation of a holomorphic eigenvalue problem is considered. The main purpose is to present a construction by which the derivation of the asymptotic error estimations for the approximate eigenvalues of Fredholm operator functions can be reduced to the derivation of these estimations for the case of matrix functions. (Some estimations for the latter problem can be derived, in one's turn, from the error estimations for the zeros of the corresponding determinants.) The asymptotic error estimations are considered in part II of this paper, in [10]. By the presented construction also the stability of the algebraic multiplicity of eigenvalues by regular approximation is proved in Section 3

The presented construction, in essence, reproduces the constructions in [7] for the case of the compact approximation in subspaces and in [9] for the case of projection—like methods. It is simpler to use than similiar construction in [8], and allows unified consideration of the general case and the case of projection—like methods, what in [8, 9] was not achieved     

Examples of non-archimedean nuclear Frécheat spaces without a Schauder basis

We solve the problem of the existence of a Schauder basis in non-archimedean Fréchet spaces of countable type (stated in [3]). Using examples of real nuclear Fréchet spaces without a Schauder basis (of Bessaga [1]), Mitiagin [5] and Vogt [10]) we construct examples of non-archimedean nuclear Fréchet spaces without a Schauder basis (even without the bounded approximation property).     

SADDLEPOINT APPROXIMATIONS TO THETWO-SAMPLE PERMUTATION TESTS

1.IntroductionPermutationtestisapowerfultoolinstatisticalinference.However,itsapplicationishinderedbythecomputationalintensity.Onewaytoovercomethisdifficultyistouseasymptoticapproximations.EdgeworthexpansionshavebeenobtainedbyAlbers,BickelandVanZ..tti]jBickelandVanZwet[ZItRobinsonlg]tJohnandR.bi....[lo]tandthesegivegoodapproximationsinmostcases.However,theseEdgeworthexpansionssometimescangiveveryinaccurateresultsinthetailsofthedistribution,withwhichwearemostconcernedforinferenceandforcon…