On the real accuracy of approximation in the central limit theorem. II

In the present article, we obtain new explicit estimates for accuracy of approximation in the central limit theorem (CLT). We construct these approximations with the use of asymptotic expansions. We compare the estimates with the real accuracy of approximation for a specific distribution. We also discuss the following question: Why the estimate from the Berry–Esseen theorem cannot catch even the order of proximity of distributions in the CLT?     

Truncation Error on Wavelet Sampling Expansions

We estimate the truncation error of sampling expansions on translationinvariant spaces, generated by integer translations of a single functionand on wavelet subspaces of L ₂(R). As a byproduct of themain result, we get the classical Jagerman's bound for Shannon's samplingexpansions. We also examine this error on certain wavelet sampling expansions.     

特殊Hermite展开的乘子定理

本文通过建立与特殊Hermite展开相对应的Littlewood-Paley分解和相关的扭曲卷积核的L2估计,得到特殊Hermite展开的乘子定理,作为该结果的应用,给出了Hermite函数及Laguerre函数展开的乘子定理。     

Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method

By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported. The first author was supported by China Postdoctoral Sciences Foundation.     

GEOMETRY OF EXPONENTIAL TYPE REGRESSION MODELS AND ITS ASYMPTOTIC INFERENCE

In this paper,exponential type regression modeis are considered from geometric point of view.The stochastic expansions relsting to the estimate are derived and are used to study several asymptotic inference problems.The biases and the covariances relating to the estimate may be calculated based on the expansions.The information loss of the estimate and a limit theorem connected with the observed and expected Fisher informations are obtained in terms of the curvatures.     

On the application of the method of matching asymptotic expansions to a singular system of ordinary differential equations with a small parameter

We consider a bisingular initial value problem for a system of ordinary differential equations with a single small parameter, the asymptotics of whose solution can be constructed in the form of power-logarithmic series on several boundary layers and an external layer. To use the method of matching asymptotic expansions, we prove theorems that permit one to make the passage between two adjacent layers and obtain a uniform estimate of the approximation to the solution by a composite asymptotic expansion.     

Multiplier theorems for special Hermite expansions on Cn

The weak type (1,1) estimate for special Hermite expansions on Cn is proved by using the Calderón-Zygmund decomposition. Then the multiplier theorem in Lp(1相似文献   

Regularization of the Cauchy problem for the Laplace equation

We suggest a method for regularizing the solution of the Cauchy problem for the Laplace equation by introducing the biharmonic operator with a small parameter. We show that if there exists a solution of the original problem, then the difference between the spectral expansions of solutions of the original and regularized equations tends to zero in the space of square integrable functions as the regularization parameter tends to zero. If the original solution belongs to a Sobolev class, then we use results of Il’in’s spectral theory to derive an estimate for the rate of the convergence of the regularized solution to the exact solution.     

ON THE DEGREE OF APPROXIMATION BY WAVELET EXPANSIONS

In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x) ∈ L2 continuous in a finite interval (a,b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.     

Asymptotic properties of integrated square error and cross-validation for kernel estimation of a regression function

Summary We derive an asymptotic expansion of integrated square error in kernel-type nonparametric regression. A similar result is obtained for a cross-validatory estimate of integrated square error. Together these expansions show that cross-validation is asymptotically optimal in a certain sense.     

On the distribution of the coefficients of normal forms for Frobenius expansions

Frobenius expansions are representations of integers to an algebraic base which are sometimes useful for efficient (hyper)elliptic curve cryptography. The normal form of a Frobenius expansion is the polynomial with integer coefficients obtained by reducing a Frobenius expansion modulo the characteristic polynomial of Frobenius. We consider the distribution of the coefficients of reductions of Frobenius expansions and non-adjacent forms of Frobenius expansions (NAFs) to normal form. We give asymptotic bounds on the coefficients which improve on naive bounds, for both genus one and genus two. We also discuss the non-uniformity of the distribution of the coefficients (assuming a uniform distribution for Frobenius expansions).     

Random Homeomorphisms and Fourier Expansions

We prove that a random change of variable in general improves convergence properties of the Fourier expansions, and we give a precise quantitative estimate of the phenomenon. Submitted: December 1997, final version: May 1998     

Richardson extrapolation of iterated discrete projection methods for eigenvalue approximation

In this paper, the eigenvalue approximation of a compact integral operator with a smooth kernel is discussed. We propose asymptotic error expansions of the iterated discrete Galerkin and iterated discrete collocation methods, and asymptotic error expansion of approximate eigenvalues. We then apply Richardson extrapolation to obtain higher order super-convergence of eigenvalue approximations. Numerical examples are presented to illustrate the theoretical estimate.     

On the spectral problem arising in the solution of a mixed problem for the heat equation with a mixed derivative in the boundary condition

We study issues related to the uniform convergence of the Fourier series expansions of Hölder class functions in the system of eigenfunctions corresponding to a spectral problem obtained from a mixed problem for the heat equation. We prove a theorem on the equiconvergence of these expansions with expansions in a well-known orthonormal basis.     

Multiplier theorems for special Hermite expansions on C n

The weak type (1,1) estimate for special Hermite expansions on Cⁿ is proved by using the Calder/'on-Zygmund decomposition. Then the multiplier theorem in L^p(lpα) is obtained. The special Mermite expansions in twisted Hardy space are also considered. As an application, the multipliers for a certain kind of Laguerre expansions are given in L^p space.     

Multiplier theorems for special Hermite expansions on Cn

The weak type (1,1) estimate for special Hermite expansions on Cⁿ is proved by using the Calder/'on-Zygmund decomposition. Then the multiplier theorem in L^p(lpα) is obtained. The special Mermite expansions in twisted Hardy space are also considered. As an application, the multipliers for a certain kind of Laguerre expansions are given in L^p space.     

A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of (□−λ)u=f and to obtain a uniform Lp estimate for a family of bump functions traveling to infinity.     

On the equisummability of Hermite and Fourier expansions

We prove an equisummability result for the Fourier expansions and Hermite expansions as well as special Hermite expansions. We also prove the uniform boundedness of the Bochner-Riesz means associated to the Hermite expansions for polyradial functions.     

Metric Properties and Exceptional Sets of the Oppenheim Expansions over the Field of Laurent Series

We investigate metric properties of the polynomial digits occurring in a large class of Oppenheim expansions of Laurent series, including Lüroth, Engel, and Sylvester expansions of Laurent series and Cantor infinite products of Laurent series. The obtained results cover those for special cases of Lüroth and Engel expansions obtained by Grabner, A. Knopfmacher, and J. Knopfmacher. Our results applied in the cases of Sylvester expansions and Cantor infinite products are original. We also calculate the Hausdorff dimensions of different exceptional sets on which the above-mentioned metric properties fail to hold.     

Estimating output gains by means of Luenberger efficiency measures

In this paper we propose a new measure of input allocative efficiency that we estimate using directional distance functions. Our new measure compares the gain in output if a firm reduces technical inefficiency for the direct production possibility set and the gain in output if the firm reduces technical inefficiency for the indirect production possibility set. Because the directional distance function uses a translated origin, the gain in output from an optimal reallocation of inputs can be estimated for non-radial expansions in output. We estimate efficiency for Japanese banks during 1992–1999. The gains in outputs from reducing allocative inefficiency by reallocating inputs are greater than the gains in outputs that can be attained by reducing technical inefficiency.