Search

In the literature, most known sequence transformations can be written as a ratio of two determinants. But, it is not always this case. One exception is that the sequence transformation proposed by Brezinski, Durbin, and Redivo-Zaglia cannot be expressed as a ratio of two determinants. Motivated by this, we will introduce a new algebraic tool—pfaffians, instead of determinants in the paper. It turns out that Brezinski-Durbin-Redivo-Zaglia’s transformation can be expressed as a ratio of two pfaffians. To the best of our knowledge, this is the first time to introduce pfaffians in the expressions of sequence transformations. Furthermore, an extended transformation of high order is presented in terms of pfaffians and a new convergence acceleration algorithm for implementing the transformation is constructed. Then, the Lax pair of the recursive algorithm is obtained which implies that the algorithm is integrable. Numerical examples with applications of the algorithm are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied. [ABSTRACT FROM AUTHOR]

The univariate spline quasi-interpolants (QIs) studied in this paper are approximation operators using B-spline expansions with coefficients that are linear combinations of discrete values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find QIs that have both good approximation orders and uniform norms which are bounded independently of the given partition. Near-best QIs are obtained by minimizing an upper bound for the infinity norm of QIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of some families of near-best discrete quasi-interpolants (dQIs) of approximation order 3. [ABSTRACT FROM AUTHOR]

Very recently Deo, in the paper “Simultaneous approximation by Lupas operators with weighted function of Szasz operators” [J. Inequal. Pure Appl. Math., 5, No. 4 (2004)] claimed to introduce the integral modifications of Lupas operators. These operators were first introduced in 1993 by Gupta and Srivastava. They estimated the simultaneous approximation for these operators and called them Baskakov-Szasz operators. There are several misprints in the paper by Deo. This motivated us to perform subsequent investigations in this direction. We extend the study and estimate a saturation result in simultaneous approximation for the linear combinations of these summation-integral-type operators. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of fast and efficient indexing techniques for sequences evolving in non-Euclidean spaces. This problem has several applications in the areas of human activity analysis, where there is a need to perform fast search, and recognition in very high dimensional spaces. The problem is made more challenging when representations such as landmarks, contours, and human skeletons etc. are naturally studied in a non-Euclidean setting where even simple operations are much more computationally intensive than their Euclidean counterparts. We propose a geometry and data adaptive symbolic framework that is shown to enable the deployment of fast and accurate algorithms for activity recognition, dynamic texture recognition, motif discovery. Toward this end, we present generalizations of key concepts of piece-wise aggregation and symbolic approximation for the case of non-Euclidean manifolds. We show that one can replace expensive geodesic computations with much faster symbolic computations with little loss of accuracy in activity recognition and discovery applications. The framework is general enough to work across both Euclidean and non-Euclidean spaces, depending on appropriate feature representations without compromising on the ultra-low bandwidth, high speed and high accuracy. The proposed methods are ideally suited for real-time systems and low complexity scenarios. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [ Cîndea, Fernández-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013], a so called primal method is described leading to a strongly convergent approximation of boundary controls: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, we study the performance of static solutions for two-stage adjustable robust linear optimization problems with uncertain constraint and objective coefficients and give a tight characterization of the adaptivity gap. Computing an optimal adjustable robust optimization problem is often intractable since it requires to compute a solution for all possible realizations of uncertain parameters (Feige et al. in Lect Notes Comput Sci 4513:439-453, ). On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently for most dynamic optimization problems. We show that for a fairly general class of uncertainty sets, a static solution is optimal for the two-stage adjustable robust linear packing problems. This is highly surprising in view of the usual perception about the conservativeness of static solutions. Furthermore, when a static solution is not optimal for the adjustable robust problem, we give a tight approximation bound on the performance of the static solution that is related to a measure of non-convexity of a transformation of the uncertainty set. We also show that our bound is at least as good (and in many case significantly better) as the bound given by the symmetry of the uncertainty set (Bertsimas and Goyal in Math Methods Oper Res 77(3):323-343, ; Bertsimas et al. in Math Oper Res 36(1):24-54, ). [ABSTRACT FROM AUTHOR]

Recently, Lal (Appl. Math. Comput. 209:346-350, 2009) has determined the degree of approximation of a function belonging to Lip α and weighted [InlineEquation not available: see fulltext.]-classes using product [InlineEquation not available: see fulltext.] summability with non-increasing weights [InlineEquation not available: see fulltext.]. In this paper, we determine the degree of approximation of function [InlineEquation not available: see fulltext.], conjugate to a 2 π-periodic function f belonging to weighted [InlineEquation not available: see fulltext.]-class by dropping the monotonicity on the generating sequence [InlineEquation not available: see fulltext.] with a new (proper) set of conditions, which in turn generalizes the results of Mishra et al. (Bull. Math. Anal. Appl., 2013) on [InlineEquation not available: see fulltext.]-class and rectifies (removes) the errors of Mishra et al. (Mat. Vesn., 2013). Few examples and applications are also highlighted in this manuscript. MSC: Primary 42B05, 42B08, 40G05, 41A10. [ABSTRACT FROM AUTHOR]

In this paper we propose a Fully Polynomial Time Approximation Scheme for a class of optimization problems where the feasible region is a polyhedral one and the objective function is the sum or product of ratios of linear functions. The class includes the well known ones of Linear (Sum-of-Ratios) Fractional Programming and Multiplicative Programming. [ABSTRACT FROM AUTHOR]

The extension of rough set model is an important research direction in the rough set theory. In this paper, based on the rough set model over two universes, we firstly propose the variable precision rough set model (VPRS-model) over two universes using the inclsion degree. Meantime, the concepts of the reverse lower and upper approximation operators are presented. Afterwards, the properties of the approximation operators are studied. Finally, the approximation operators with two parameters are introduced as a generalization of the VPRS-model over two universes, and the related conclusions are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm L p (1 ≤ p ≤ ∞) of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results. [ABSTRACT FROM AUTHOR]

We study d-variate L2-approximation for a weighted unanchored Sobolev space having smoothness m≥1. This space is equipped with an unusual norm which is, however, equivalent to the norm of the d-fold tensor product of the standard Sobolev space. One might hope that the problem should become easier as its smoothness increases. This is true for our problem if we are only concerned with asymptotic analysis: the nth minimal error is of order n−( m− δ) for any δ>0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? It is easy to prove that no matter how the weights are chosen, the waiting period is at least m d, even if the error demand ε is arbitrarily close to 1. Hence, for m≥2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So we will be unable to vanquish the curse of dimensionality unless m=1, i.e., unless the smoothness is minimal. In this paper, we prove the more difficult fact that our problem can be tractable if m=1. That is, we can find an ε-approximation using polynomially-many (in d and ε−1) information operations, even if only function values are permitted. When m=1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε−1) information operations, independently of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d. [ABSTRACT FROM AUTHOR]

Abstract  In this paper we present a complete classification of the isolated central configurations of the five-body problem with equal masses. This is accomplished by using the polyhedral homotopy method to approximate all the isolated solutions of the Albouy-Chenciner equations. The existence of exact solutions, in a neighborhood of the approximated ones, is then verified using the Krawczyk method. Although the Albouy-Chenciner equations for the five-body problem are huge, it is possible to solve them in a reasonable amount of time. [ABSTRACT FROM AUTHOR]

In this paper we study zero-sum stochastic games. The optimality criterion is the long-run expected average criterion, and the payoff function may have neither upper nor lower bounds. We give a new set of conditions for the existence of a value and a pair of optimal stationary strategies. Our conditions are slightly weaker than those in the previous literature, and some new sufficient conditions for the existence of a pair of optimal stationary strategies are imposed on the primitive data of the model. Our results are illustrated with a queueing system, for which our conditions are satisfied but some of the conditions in some previous literatures fail to hold. [ABSTRACT FROM AUTHOR]

In this paper, we prove a stability result for measure perturbations of some class of stationary distributions of a Vlasov equation. We use this result to prove that the N particles approximation of these stationary distributions is uniformly valid on a time scale of order N 1/8, which is much longer than the usual log N scale. We also prove similar results for the approximation of the two-dimensional Euler equation by the vortex blob method. [ABSTRACT FROM AUTHOR]

This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E| h( X 1, X 2)|2 < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence. [ABSTRACT FROM AUTHOR]

In [9], we proved numerically that spaces generated by linear combinations of some two-dimensional Haar functions exhibit unexpectedly nice orders of approximation for solutions of the single-layer potential equation in a rectangle. This phenomenon is closely related, on the one hand, to the properties of the approximation method of hyperbolic crosses and on the other to the existence of a strong singularity for solutions of such boundary integral equations. In the present paper, we establish several results on the approximation for the hyperbolic crosses and on the best N-term approximations by linear combinations of Haar functions in the H s -norms, −1 < s < 1/2; this provides a theoretical base for our numerical research. To the author's best knowledge, the negative smoothness case s < 0 was not studied earlier. [ABSTRACT FROM AUTHOR]

In this paper, the problem of an exponential stability for time-delay systems with interval time-varying delays and nonlinear perturbations is investigated. Based on the Lyapunov method, a new delay-dependent criterion for exponential stability is established in terms of LMI (linear matrix inequalities). Numerical examples are carried out to support the effectiveness of our results. [ABSTRACT FROM AUTHOR]

This paper brings together two fundamental topics: polyhedral projection and parametric linear programming. First, it is shown that, given a parametric linear program (PLP), a polyhedron exists whose projection provides the solution to the PLP. Second, the converse is tackled and it is shown how to formulate a PLP whose solution is the projection of an appropriately defined polyhedron described as the intersection of a finite number of halfspaces. The input to one operation can be converted to an input of the other operation and the resulting output can be converted back to the desired form in polynomial time-this implies that algorithms for computing projections or methods for solving parametric linear programs can be applied to either problem class. [ABSTRACT FROM AUTHOR]

A robust reliable control with integral quadratic constraint (IQC) performance for a class of uncertain systems with state and input delays is considered in this paper. Two classes of failure situations for sensor or actuator are studied. In the first class, a delay-dependent criterion for time-delay systems without perturbations is proposed to design the reliable control with IQC performance. Next, a criterion for uncertain time-delay systems with parameter uncertainties is obtained via simple derivations. The linear matrix inequality (LMI) approach is used to design a robust reliable state feedback control with IQC performance. In the second class, a reliable control with IQC performance is also provided from he previous method. A numerical example is given to illustrate the effectiveness of the procedure. [ABSTRACT FROM AUTHOR]

The main result of this paper is a generalized Stieltjes criterion for the complete indeterminacy of interpolation problems in the Stieltjes class. This criterion is a generalization to limit interpolation problems of the classical Stieltjes criterion for the indeterminacy of moment problems. It is stated in terms of the Stieltjes parameters M j and N j . We obtain explicit formulas for the Stieltjes parameters. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class. [ABSTRACT FROM AUTHOR]

In this paper, the variations of both subunitary polar factors and Hermitian positive semidefinite polar factors in the polar decomposition are studied. New perturbation bounds of both polar factors are given without the restriction that A and its perturbed matrix à have the same rank. These improve recent results. [ABSTRACT FROM AUTHOR]

The paper considers the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, the author proves necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided, infinite block-band matrix composed with respect to integrals over intervals of length h > 0 of the matrix of the system of differential equations. Also, the author constructs a family of two-point boundary-value problems on a finite interval that approximates the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide an error analysis for the multicategory support vector machine (MSVM) classificaton problems. We establish the uniform convergency approach for MSVMs and estimate the misclassification error. The main difficulty we overcome here is to bound the offset vector. As a result, we confirm that the MSVM classification algorithm with polynomial kernels is always efficient when the degree of the kernel polynomial is large enough. Finally the rate of convergence and examples are given to demonstrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider algorithms for constructing n-terms approximations with nonnegative coefficients. The convergence theorem is proved for a “positive” analog of the Pure Greedy Algorithm. We establish a condition on the sequence of weakness coefficients which is sufficient for the convergence of the Positive Weak Greedy Algorithm. This condition is also necessary for the class of monotone sequences. [ABSTRACT FROM AUTHOR]

Abstract  In this paper we consider the problem of the approximation of the integral of a smooth enough function f(x,y) on the standard simplex $${\int\limits_{\Delta _{2} } {f{\left( {x,y} \right)}dxdy} } = {\sum\limits_{\alpha = 1}^3 {{\sum\limits_{i,j} {A_{{\alpha ij}} \frac{{\alpha ^{{i j}} }}{{\alpha x^{i} \alpha y^{j} }}f{\left( {x_{\alpha } ,y_{\alpha } } \right)} E{\left( f \right)}} }} }$$ where the nodes are the vertices of the simplex. Such kind of quadratures belong to a more general class of formulas for numerical integration, which are called boundary-type quadrature formulas. We discuss three classes of such formulas that are exact for algebraic polynomials and generate embedded pairs. We give bounds for the truncation errors and conditions for convergence. Finally, we show how to organize an algorithm for the automatic computation of the quadratures with estimate of the errors and provide some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator φ( hA) v, φ ( z) = ( e z − 1)/ z. In this way, it is possible to approximate φ ( hA) v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to “higher” order φ k functions, k≥0. [ABSTRACT FROM AUTHOR]

In the paper, a generalization of a known theorem by Hardy and Young is obtained; a formula interrelating the integral of a 2π-periodic function over the period with the integral over the entire axis is established; new approximation characteristics for functions belonging to saturation classes of continuity modules of different orders for the spaces Lp of periodic functions are provided, and some issues concerning approximation, in the uniform metric, of continuous periodic functions even with respect to each of their variables and having nonnegative Fourier coefficients are considered. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators. [ABSTRACT FROM AUTHOR]

The study of Tchebycheff spaces (generalizing the space of algebraic polynomials) and extremal problems related to them began one and a half centuries ago. Recently, many facts of approximation theory have been understood and reinterpreted from the point of view of general principles of the theory of extremum and convex duality. This approach not only allowed one to prove the previously known results for algebraic polynomials and generalized polynomials in a unified way, but also enabled one to obtain new results. In this paper, we work out this direction with special attention to the optimal recovery problems. [ABSTRACT FROM AUTHOR]

In this paper we propose a technique of approximation for the generalized Riemann-Stieltjes integral and we found an analogue for Newton-Cotes formulas in the case n = 2 and n = 3. [ABSTRACT FROM AUTHOR]

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in L p generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp. [ABSTRACT FROM AUTHOR]

In this paper we analyze the behavior of the solution of the dissipative Boussinesq systems where α, β, c > 0 are parameters. Those systems model two-dimensional small amplitude long wavelength water waves. For α ≤ 1, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every β, c and almost every α ≤ 1, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form. [ABSTRACT FROM AUTHOR]

The study of zeros of orthogonal functions is an important topic. In this paper, by improving the middle variable x(t), we've got a new form of asymptotic approximation, completed with error bounds, it is constructed for the Jacobi functions ϕμ(αβ) (t) (α > -1) as μ → + ∞. Besides, an accurate approximation with error bounds is also constructed correspondingly for the zeros tμ,8 of ϕμ(αβ) (t) (α ≥ -0) as μ → + ∞, uniformly with respect to s = 1,2,.... [ABSTRACT FROM AUTHOR]

In this paper, a new method for the approximate solution of linear singular integral equations defined on smooth closed curves is proposed and justified. [ABSTRACT FROM AUTHOR]

In this paper, large commutative subalgebras of quantum algebras (in particular, quantum polynomial, matrix, and Weyl algebras) are studied. The notions of the transcendence degree and the Krull dimension are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush-Kuhn-Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn. [ABSTRACT FROM AUTHOR]

In this paper, we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We discuss also the consistency and rate of convergence of sample average approximations of such stochastic problems. [ABSTRACT FROM AUTHOR]

The paper deals with approximation of a continuous function, on a finite interval, which changes convexity finitely many times, by algebraic polynomials which are coconvex with it. We give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian-Totik moduli of smoothness. [ABSTRACT FROM AUTHOR]

In this paper, we solve the problem of the exact order of decrease of uniform moduli of smoothness for the classes of 2π-periodic functions of several variables with a given majorant of the sequence of total best approximations in the metric of L p, 1 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

Letbe a unit sphere of thed-dimensional Euclidean space Rd and let(0 < p= 1) denote the real Hardy space onFor 0 < p= 1 andletEj(f,Hp) (j= 0, 1, ...) be the best approximation offby spherical polynomials of degree less than or equal toj, in the spaceGiven a distributionfonits Cesàro mean of order d>-1 is denoted byFor 0

In this paper we propose and study an iterative method for solving a BVP for a triharmonic type equation. It is based on using a boundary-domain operator defined on pairs of boundary and domain functions in combination with parametric extrapolation technique. This method iteratively reduces the BVP for sixth order equation to a sequence of BVPs for Poisson equation. [ABSTRACT FROM AUTHOR]

Results on the asymptotic analysis of a model of autoresonance using a natural small parameter, the amplitude of forcing oscillations, are presented. The given forcing perturbation is rapidly oscillating oscillations with small amplitude and slow varying frequency. The goal of the paper is to reveal conditions under which the trajectory of the system goes away from the initial equilibrium point by a distance of order 1 under the influence of such a perturbation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate both combinatorial and complexity aspects of the problem of finding best transitive approximations to simple graphs. These problems are addressed in an interlocked way. We provide new and simple proofs of known results and in addition prove some new theorems. [ABSTRACT FROM AUTHOR]

In this paper we present two approximation algorithms for the permutation flow shop problem with makespan objective. The first algorithm has an absolute performance guarantee, the second one is an $$O(\sqrt {m{\text{ log }}m} )$$ -approximation algorithm. The last result is almost best possible approximation algorithm which can be obtained using the trivial lower bound (maximum of the maximum machine load and the maximum job length) (Potts, Shmoys, and Williamson, 1991). [ABSTRACT FROM AUTHOR]

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 Garg’s 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. [ABSTRACT FROM AUTHOR]

The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in Lp spaces for 1 < p < ∞ and proves that if f(x)∈Lp[-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈ Rnl such that ∥f(x)-r(x)∥ Lp ≤;Cp,l,bω (f,n-1)Lp, where Rnl indicates all rational functions whose denominators consist of polynomials of degree n and numerators polynomials of degree l, and Cp, l,b is a positive constant depending only on p, l and b which relates to the distance among the sign change points of f(x) and will be given in §3. [ABSTRACT FROM AUTHOR]

Let f be an entire function of finite type with respect to finite order p in Cn and let E be a subset of an open cone in a certain n-dimensional subspace R2n (= Cn) (the smaller p, the sparser E). We assume that this cone contains a ray {z = tz0 ∈ Cn: t>0}. It is shown that the radial indicator hf (z0) of f at any point z0 ∈ Cn \ {0} may be evaluated in terms of function values at points of the discrete subset E. Moreover, if f tends to zero fast enough as z → ∞ over E, then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on p and E, which are close to exact conditions, the function f bounded on E is bounded on the ray. [ABSTRACT FROM AUTHOR]

This paper is motivated by S. Park [10] in which the injective cover of left R[x]-module M[x[sup -1]] of inverse polynomials over a left R-module M was discussed. The author considers the Ω-covers of modules and shows that if η: P → M is an Ω-cover of M, then [η[sup S,≤]] : [P[sup S,≤]] → [M[sup S,≤]] is an [Ω[sup S,≤]]-cover of left [[R[sup S,≤]]]-module [M[sup S,≤]], where Ω is a class of left R-modules and [M[sup S,≤]] is the left [[R[sup S,≤]]]-module of generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[R[sup S,≤]]]-module [M[sup S,≤]] are discussed. [ABSTRACT FROM AUTHOR]