Generalized Strang–Fix condition for scattered data quasi-interpolation

Quasi-interpolation is very useful in the study of the approximation theory and its applications, since the method can yield solutions directly and does not require solving any linear system of equations. However, quasi-interpolation is usually discussed only for gridded data in the literature. In this paper we shall introduce a generalized Strang–Fix condition, which is related to nonstationary quasi-interpolation. Based on the discussion of the generalized Strang–Fix condition we shall generalize our quasi-interpolation scheme for multivariate scattered data, too. AMS subject classification 41A63, 41A25, 65D10Zong Min Wu: Supported by NSFC No. 19971017 and NOYG No. 10125102.     

Upper Estimates in Direct Inequalities for Bernstein-Type Operators

We obtain explicit upper estimates in direct inequalities with respect to the usual sup-norm distance for Bernstein-type operators. Our approach combines analytical and probabilistic techniques based on representations of the operators in terms of stochastic processes. We illustrate our results by considering some classical families of operators, such as Weierstrass, Szàsz, and Bernstein operators.     

Global smoothness and uniform convergence of smooth Gauss–Weierstrass singular operators

In this article we continue with the study of smooth Gauss–Weierstrass singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L_p norm, 1≤p≤∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.     

Some Shift-Invariant Integral Operators, Univariate Case, Revisited

In recent articles the first author and H. Gonska [e.g., see G. Anastassiou, C. Cottin, and H. Gonska, Global smoothness of approximating functions, Analysis, 11, 43–57 (1991); G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] studied global smoothness preservation by some univariate and multivariate linear operators over compact domains and ⁿ , n 1. In particular, they studied a very general positive linear integral type operator [e.g., see G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] over ⁿ that was introduced through a convolution-like integration of another general positive linear operator with a scaling-type function. In this article the authors, among others, extend and generalize [G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)]. Also certain new similar but more general integral operators are introduced and studied. These operators arise in a natural way, and for all these sufficient conditions are given for shift invariance, preservation of higher-order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preservation, and preservation of continuous probabilistic distribution functions. Several examples of very general specialized operators, old and new, are given that satisfy all the above properties.     

The Inverse Spectral Problem for the Sturm-Liouville Operators with Discontinuous Coefficients

We study the inverse spectral problem for the Sturm–Liouville operator whose piecewise constant coefficient A(x) has discontinuity points x _k , k=1,...,n, and jumps A _k =A(x _k +0)/A(x _k -0). We show that if the discontinuity points x ₁,...,x _n are noncommensurable, i.e., none of their linear combinations with integer coefficients vanishes; then the spectral function of the operator determines all discontinuity points x _k and jumps A _k uniquely. We give an algorithm for finding x _k and A _k in finitely many steps.     

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation.     

The Characterization of the Derivatives for Linear Combinations of Post–Widder Operators inLp

The aim of the paper is to characterize the global rate of approximation of derivativesf^(l)through corresponding derivatives of linear combinations of Post–Widder operators in an appropriate weightedL_p-metric using a weighted Ditzian and Totik modulus of smoothness, and also to characterize derivatives of these operators in Besov spaces of Ditzian–Totik type.     

On some families of multi-point iterative methods for solving nonlinear equations

Some semi-discrete analogous of well known one-point family of iterative methods for solving nonlinear scalar equations dependent on an arbitrary constant are proposed. The new families give multi-point iterative processes with the same or higher order of convergence. The convergence analysis and numerical examples are presented.     

On boundary-value problem for hyperbolic-type equation of the third order

We formulate some boundary-value problems for a linear third-order equation with hyperbolic operator in the main part and study the unique solvability. Under certain conditions to given functions, using the Riemann method, we obtain an integral representation of solutions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 591–603, October–December, 2007.     

基于POD-RBF方法的管道内壁几何识别

针对天然气、石油等管道内部被腐蚀问题,基于本征正交分解-径向基函数(POD-RBF)提出了一种管道内壁几何识别方法. 考虑静磁场并建立管道的简化有限元模型,构建变几何样本库,实现了POD-RBF对任意形状的响应预测. 该方法在降阶分析的同时避免了迭代过程中因几何的改变需反复求解刚度矩阵,在很大程度上提高了计算效率. 采用灰狼优化(GWO)算法对目标函数实施优化,避免了在变几何过程中灵敏度的求解. 算例结果显示,该文方法可高效准确地反演管道内壁的几何形状,即使在引入噪声后GWO算法仍具有较好的稳定性.     

Old and new formulas for the Hopf–Stiefel and related functions

We give a list of the various formulas describing the Hopf–Stiefel function and provide direct, arithmetical proofs of their equivalence.     

Sufficient Conditions of Isolated Minimizers for Constrained Programming Problems

In this article, sufficient optimality conditions for nonsmooth programming problems with inequality constraints are studied. When the objective and constraint functions are ?-stable at some point, a first-order sufficient optimality condition for an isolated local minimizer of order 1 is established. A second-order sufficient optimality condition for an isolated local minimizer of order 2 is obtained in terms of the lower Dini second-order directional derivatives of the Lagrangian function. The obtained results extend and improve the ones found by Ward [21 D.E. Ward ( 1994 ). Characterizations of strict local minima and nacessary conditions for weak sharp minima . Journal of Optimization Theory and Applications 80 : 551 – 571 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Maximal approximation order for a box-spline semi-cardinal interpolation scheme on the three-direction mesh

Let M be the centred 3-direction box-spline whose direction matrix has every multiplicity 2. A new scheme is proposed for interpolation at the vertices of a semi-plane lattice from a subspace of the cardinal box-spline space generated by M. The elements of this semi-cardinal box-spline subspace satisfy certain boundary conditions extending the not-a-knot end-conditions of univariate cubic spline interpolation. It is proved that the new semi-cardinal interpolation scheme attains the maximal approximation order 4. AMS subject classification 41A15, 41A05, 41A25, 41A63, 39A70, 47B35     

Gröbner–Shirshov Bases for Universal Enveloping Conformal Algebras of Simple Conformal Lie Superalgebras of Type WN

For simple conformal Lie superalgebras of type W_N, Gröbner–Shirshov bases of their universal enveloping associative conformal algebras are found. The universal enveloping algebras considered correspond to a minimal locality function for which there is an injective embedding.