Derivation and validation of initial-value methods for boundary-value problems for difference equations

In this paper, we develop the theory of invariant imbedding for general classes of two-point boundary-value problems for difference equations. In addition to deriving invariant imbedding equations, we show that the functions satisfying these equations in fact solve the original boundary-value problems.     

确定性与随机化框架下的Sobolev类上的嵌入与积分的复杂性

本文研究各向异性Sobolev类上的嵌入以及积分问题的复杂性.我们得到这些问题在确定性、随机化框架以及平均框架下n-重最小误差的精确阶.所得结果表明在非嵌入连续函数空间情形,随机误差与平均误差实质性地小于确定性误差.从数量级看,对于嵌入问题,收敛阶最大改进可达到n-1+ε,这里ε是任意正数.对于积分问题最大改进可达到n...     

Bi-Lipschitz approximation by finite-dimensional imbeddings

Gromov’s celebrated systolic inequality from ’83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1+e){(1+\epsilon)}-bi-Lipschitz (onto its image), finite-dimensional imbedding for every ${\epsilon > 0}${\epsilon > 0}, using the first variation formula and the mean value theorem.     

The stechkin problem for partial derivation operators on classes of finitely smooth functions

In this paper we find the order of the best value of approximation by operators from one space of functions with integral norm to another such space on classes of finitely smooth functions of partial derivation operators with bounded norm. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 77–86, January, 2000.     

Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.     

Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and general-ized Sobolev classes into Lq([0,1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0,1]d)) to Lq([0,1]d) space for all 1 q,p ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.     

用指数型整函数的最佳限制逼近

我们研究了由仅有实零点的代数多项式导出的微分算子确定的广义Sobolev类利用指数型整函数作为逼近工具的最佳限制逼近问题.利用Fourier变换和周期化等方法,得到在L_2(R)范数下的广义Sobolev光滑函数类的相对平均宽度和最佳限制逼近的精确常数,以及当0是这个代数多项式的一个至多2重的零点时,得到最佳限制逼近在L_1(R)范数和一致范数下的广义Sobolev类的精确到阶的结果.     

A rational approximation and its applications to nonlinear partial differential equations on the whole line

An orthogonal system of rational functions is discussed. Some inverse inequalities, imbedding inequalities and approximation results are obtained. Two model problems are considered. The stabilities and convergences of proposed rational spectral schemes and rational pseudospectral schemes are proved. The techniques used in this paper are also applicable to other problems on the whole line. Numerical results show the efficiency of this approach.     

Error estimation of Hermite spectral method for nonlinear partial differential equations

Hermite approximation is investigated. Some inverse inequalities, imbedding inequalities and approximation results are obtained. A Hermite spectral scheme is constructed for Burgers equation. The stability and convergence of the proposed scheme are proved strictly. The techniques used in this paper are also applicable to other nonlinear problems in unbounded domains.

     

Interpolating-orthogonal wavelet systems

Based upon Meyer wavelets, new systems of periodic wavelets and wavelets on the whole axis are constructed; these systems are orthogonal and interpolating simultaneously. Estimates of the errors of approximation of different classes of smooth functions by these wavelets are obtained.     

Jensen measures and boundary values of plurisubharmonic functions

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions inB-regular domain. This theorem implies that the two classes of Jensen measures coincide inB-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above. The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.     

Some extremal properties of multivariate polynomial splines in the metric Lp(ℝd)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (?^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (?^d in the metric L_p((?^d).     

Singular numbers of the imbedding operators for certain classes of analytic and harmonic functions

One obtains formulas of the strong asymptotics of the s-numbers of the imbedding operators of the weighted classes of analytic and harmonic functions into the weighted spaces L₂ on the interior subdomains.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 56–66, 1984.The author expresses his gratitude to M. Z. Solomyak for his constant support.     

Approximation by entire functions on countable unions of segments of the real axis: 2. proof of the main theorem

In this study, we consider an approximation of entire functions of Hölder classes on a countable union of segments by entire functions of exponential type. It is essential that the approximation rate in the neighborhood of segment ends turns out to be higher in the scale that had first appeared in the theory of polynomial approximation by functions of Hölder classes on a segment and made it possible to harmonize the so-called “direct” and “inverse” theorems for that case, i.e., restore the Hölder smoothness by the rate of polynomial approximation in this scale. Approximations by entire functions on a countable union of segments have not been considered earlier. The first section of this paper presents several lemmas and formulates the main theorem. In this study, we prove this theorem on the basis of earlier given lemmas.     

Functions and Sets of Smooth Substructure: Relationships and Examples

The past decade has seen the introduction of a number of classes of nonsmooth functions possessing smooth substructure, e.g., “amenable functions”, “partly smooth functions”, and “g ° F decomposable functions”. Along with these classes a number of structural properties have been proposed, e.g., “identifiable surfaces”, “fast tracks”, and “primal-dual gradient structures”. In this paper we examine the relationships between these various classes of functions and their smooth substructures. In the convex case we show that the definitions of identifiable surfaces, fast tracks, and partly smooth functions are equivalent. In the nonconvex case we discuss when a primal-dual gradient structure or g ° F decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal.     

Approximation Methods by Regular Functions

This paper is a survey of approximation results and methods by smooth functions in Banach spaces. The topics considered in the paper are the following: approximation by polynomials by C^k-functions using the method of smooth partitions of unity, approximation by the fine topology, analytic approximation and regularization in Banach spaces using the infimal convolution method.     

Decomposition theorems and approximation by a ``floating" system of exponentials

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.

     

The Gel'fand widths of anisotropic Sobolev classes of smooth functions

In this paper, we obtain some Gel'fand widths of anisotropic Sobolev periodic classes of smooth functions, and average Gel'fand widths of anisotropic Sobolev classes of smooth functions.     

Some extremal properties of multivariate polynomial splines in the metric Lp(?d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (ℝ^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (ℝ^d in the metric L_p((ℝ^d).