Limit theorems for Fourier transforms of functionals of Gaussian sequences

Summary Limit theorems with a non-Gaussian (in fact nonstable) limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence. Analogous limit theorems are here obtained for finite Fourier transforms of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence.Research supported in part by the Office of Naval Research Contract N00014-75C-0428.     

Shift-invariant bi-inner product functionals are inner products

Bi-inner product functionals generated by a pair of Bessel sequences of L² functions are introduced. It is shown that these functionals are constant multiples of the inner products of L² and l², if and only if they are shift-invariant both in space (or time) and in phase. This result is then applied to characterize dual frames and bi-orthogonal Riesz bases of L². The research of this author was supported by NSF Grant # DMS-95-05460 and ARO Contract #DAAH-04-95-10193, This author is currently visiting the Department of statistics, Stanford University, Stanford, CA 94305. The research of this author was supported by the Texas Coordinating Board of Higher Education under Grant # 999903-109.     

Best simultaneous approximation to totally bounded sequences in Banach spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. The first author is supported in part by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 06C651); the second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 10671175, 10731060) and Program for New Century Excellent Talents in University; the third author is supported in part by Projects MTM2006-13997-C02-01 and FQM-127 of Spain     

Convergence theorems and Tauberian theorems for functions and sequences in Banach spaces and Banach lattices

We prove generalized convergence theorems and Tauberian theorems for vector-valued functions and sequences of growth order γ − 1 with γ > 0 and for positive functions and sequences in Banach lattices. Then the general results are applied to obtain some interesting particular Tauberian results for various examples of operator semigroups. Among them are mean ergodic theorems for Cesàro-mean-bounded semigroups (discrete and continuous) of operators and for semigroups of positive operators. Research supported in part by the National Science Council of Taiwan. Current address: 19-18, Higashi-hongo 2-chome, Midori-ku, 226-0002 Japan.     

Ergodicity of Jackson-type queueing networks

This paper gives a pathwise construction of Jackson-type queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMPs. The paper also provides new results on the Jackson-type networks with i.i.d. driving sequences which were studied in the past.The work of this author was supported in part by a grant from the European Commission DG XIII, under the BRA Qmips contract.The work of this author was supported by a sabbatical grant from INRIA Sophia Antipolis.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

Estimating the multiplicity of a root

Summary One point iteration functions for approximating roots of a function of a single variable and sequences generated by them are considered. Methods of estimating the multiplicity of the roots to which such sequences are converging are given. In particular, multiplicity estimators are given for Newton's method, the modified Newton's method, and Laguerre's method. It is shown that the estimators for Newton's method and Laguerre's method offer improvement over those of Rall and Dekker, respectively.This research was supported by the National Science Foundation, Grant No. NSF-DCR-14-10042.     

On the sharpness of theorems concerning zero-free regions for certain sequences of polynomials

Summary In this paper, we establish the sharpness of a theorem concerning zero-free parabolic regions for certain sequences of polynomials satisfying a three-term recurrence relation. Similarly, we establish the sharpness of a zero-free sectorial region for certain sequences of Padé approximants toe ^z .Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075     

The central limit problem for,infinite products of,and Lévy processes of renewal sequences

Summary Necessary and sufficient conditions are obtained for: (i) convergence of row products from a null triangular array of renewal sequences to a particular renewal sequence and (ii) convergence of an infinite product of renewal sequences to a non-trivial limit. These products correspond to intersections of regenerative phenomena of integers. Lévy processes of such regenerative phenomena are constructed.Research partially supported by National Science Foundation Grant MCS 78-01168     

On examples of difference operators for {0,1}-valued functions over finite sets

V.I. Arnold recently formulated a geometric concept of monads and used it to study difference operators on the sets of {0,1}-valued sequences of length n. We consider particular examples of these monads and indicate one question that arises. Partially supported by grants NWO-RFBR 047.011.2004.026 (RFBR 05-02-89000-NWO_a), RFBR SS-1972.2003.1, RFBR 05-01-02805-CNRSL_a, and RFBR 05-01-01012a.     

On a theorem of Cantor-Bernstein type for algebras

Freytes proved a theorem of Cantor-Bernstein type for algbras; he applied certain sequences of central elements of bounded lattices. The aim of the present paper is to extend the mentioned result to the case when the lattices under consideration need not be bounded; instead of sequences of central elements we deal with sequences of internal direct factors of lattices. This work was supported by Science and Technology Assistance Agency under the contract No APVT-51-032002.     

Discrete Systems and their Characteristic Spectral Functions

We define first-order discrete systems in the matrix-valued case. They are characterized by sequences of pair of matrices, called admissible sequences. We present two important examples of such sequences, called Szeg? and Nehari sequences. We introduce the characteristic spectral functions associated to a first-order system. We define in particular the scattering function, the Weyl function and the reflection coefficient function and we study the relationships between these functions. Daniel Alpay wishes to thank the Earl Katz family for endowing the chair which supported his research.     

Almost sure convergence theorems in von Neumann algebras

The subadditive sequences of operators which belong to a von Neumann algebra with a faithful normal state and a given positive linear kernel are considered. We prove the almost sure convergence in Egorov’s sense for such sequences. Partially supported by the Ministry of Science and Ministry of Absorption. Partially sponsored by a grant from the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Singular and cosingular exact sequences of quasi-Banach spaces

We give a characterization of exact sequences of Banach or quasi-Banach spaces having strictly singular quotient map or strictly cosingular embedding. We present two balance principles showing that such sequences can only exist when the sizes of the subspace and the quotient space are not too different. Received: 7 November 2005 The research was supported in part by the project MTM2004-02635.     

Interpolatory integration rules and orthogonal polynomials with varying weights

We investigate which types of asymptotic distributions can be generated by the knots of convergent sequences of interpolatory integration rules. It will turn out that the class of all possible distributions can be described exactly, and it will be shown that the zeros of polynomials that are orthogonal with respect to varying weight functions are good candidates for knots of integration rules with a prescribed asymptotic distribution.Research supported by the Deutsche Forschungsgemeinschaft (AZ: Sta 299/4-2).     

GA-convexity and its applications

This paper is concerned with some sequences involving GA-convex and GA-concave functions. By investigating the monotonicity of these sequences, we obtain a class of new inequalities, some of which extend the known ones. This work is supported by Hunan Provincial Natural Science Foundation of China (No. 11JJ3011)     

Cyclicity of coefficient multipliers: Linear structure

We characterize various kinds of cyclicity of sequences of coefficient multipliers, which are operators defined on spaces of holomorphic functions. In the case of a single coefficient multiplier we characterize its cyclicity, which contrasts with the fact that such operators are never supercyclic. Moreover, it is proved that for each cyclic function there is a dense part of the linear span of its orbit all of whose vectors are cyclic. The authors have been partially supported by MCYT Grants BFM2003-03893-C02-01, MTM2004-21420-E and Plan Andaluz de Investigación Junta de Andalucía FQM-127.     

Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result.The first author was partially supported by the Russian Foundation for Basic Research Grant 02-01-00334. The second author was partially supported by the Russian Foundation for Basic Research Grant 00-01-00869. The third author was partially supported by Fundacao para a Ciencia e Tecnologia and by INVOTAN Grant.     

An exact penalty function algorithm for optimal control problems with control and terminal equality constraints,part 2

In Part 1 of this paper, implementable and conceptual versions of an algorithm for optimal control problems with control constraints and terminal equality constraints were presented. It was shown that anyL accumulation points of control sequences generated by the algorithms satisfy necessary conditions of optimality. Since such accumulation points need not exist, it is shown in this paper that control sequences generated by the algorithms always have accumulation points in the sense of control measure, and these accumulation points satisfy optimality conditions for the corresponding relaxed control problem.This work was supported by the United Kingdom Science Research Council, by the US Army Research Office, Contract No. DAA-29-73-C-0025, and by the National Science Foundation, Grant No. ENG-73-08214-A01.     

Barycentric scenario trees in convex multistage stochastic programming

This work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support. Based on earlier results, sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem. Error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation. Convergence of the approximate solutions is proven under the stated assumptions. Preliminary computational results are outlined. This work has been supported by Schweizerischen Nationalfonds Grant Nr. 21-39 575.93.