Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Let there be given a probability measure μ on the unit circle of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r_k}_k, for a certain order of the , by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψ_k}_k. We show that the matrix representation with respect to {ψ_k}_k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r_k}_k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.     

Strong approximation of Fourier series and embedding theorems

Summary In this paper, embedding results are considered which arise in the strong approximation by Fourier series. We prove several theorems on the interrelation between the classes W^r H_β^ω and H (λ,p,r,ω), the latter being defined by L. Leindler. Previous related results in Leindler’s book [2] and paper [5] are particular cases of our results.     

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.     

Linear homeomorphisms of non-classical Hilbert spaces

Let K be a complete infinite rank valued field. In [4] we studied Norm Hilbert Spaces (NHS) over K i.e. K-Banach spaces for which closed subspaces admit projections of norm ≤ 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms are bijective, no NHS is linearly homeomorphic to a proper subspace (Theorem 3.7), each operator can be approximated, uniformly on bounded sets, by finite rank operators (Theorem 3.8). These properties together — in real or complex theory shared only by finite-dimensional spaces — show that NHS are more ‘rigid’ than classical Hilbert spaces.     

Non-parametric efficiency, progress and regress measures for panel data: Methodological aspects

This purely methodological paper deals with the rôle of time in non-parametric efficiency analysis. Using both FDH and DEA technologies, it first shows how each observation in a panel can be characterized in efficiency terms vis-à-vis three different kinds of frontiers: (i) ‘contemporaneous’, (ii) ‘sequential’, and (iii) ‘intertemporal’. These are then compared with window analysis. Next, frontier shifts ‘outward’ and ‘inward’, interpreted as progress or regress are considered for the two kinds of technologies, and computational methods are described in detail for evaluating such shifts in either case. These are also contrasted with what is measured by the ‘Malmquist’ productivity index. Finally, an alternative way of identifying progress and regress, independent of the frontier notion and referring instead to some ‘benchmark’ notion, is extended here to panel data.     

Browder’s Theorems through Localized SVEP

A bounded linear operator T ∈ L(X) on aBanach space X is said to satisfy “Browder’s theorem” if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy “a-Browder’s theorem” if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T. In the last part we shall give some characterizations of operators for which “Weyl’s theorem” holds.     

Path-dependent processes and the emergence of macro-structure

Path-dependent systems of the ‘autocatalytic’ or self-reinforcing type typically possess a multiplicity of possible asymptotic outcomes or structures, with early random fluctuations determining which structure is ‘selected’.We explore a wide class of such systems, which we call non-linear Polya systems, where increments to proportions or concentrations occur with probabilities that are non-linear functions of present proportions or concentrations. We show that such systems converge to outcomes (proportions or concentrations) that are represented by the stable fixed points of these functions. These limit theorems are strong laws of large numbers for path-dependent increments, and as such they generalize the standard Borel strong law for independent increments. They are powerful and easy to use.We show applications in chemical kinetics, industrial location theory and in the emergence of technological structure in the economy.     

An asymptotic expansion for the error in a linear map that reproduces polynomials of a certain order

Han's ‘multinode higher-order expansion’ in [H] is shown to be a special case of an asymptotic error expansion available for any bounded linear map on C([a..b]) that reproduces polynomials of a certain order. The key is the formula for the divided difference at a sequence containing just two distinct points.     

Cyclic permutations and evolutionary trees

Given a tree with leaf set X, there are certain ways of arranging the elements of X in a circular order so that can be embedded in the plane and ‘preserve’ this ordering. We investigate some new combinatorial properties of these ‘circular orderings.’ We then use these properties to establish two results concerning dissimilarity maps on X that are induced by edge-weighted trees with leaf set X.     

The Heegaard Genus of Amalgamated 3-Manifolds

Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.     

Linguistic modelling and information coarsening based on prototype theory and label semantics

The theory of prototypes provides a new semantic interpretation of vague concepts. In particular, the calculus derived from this interpretation results in the same calculus as label semantics proposed by Lawry. In the theory of prototypes, each basic linguistic label L has the form ‘about P’, where P is a set of prototypes of L and the neighborhood size of the underlying concept is described by the word ‘about’ which represents a probability density function δ on [0,+∞). In this paper we propose an approach to vague information coarsening based on the theory of prototypes. Moreover, we propose a framework for linguistic modelling within the theory of prototypes, in which the rules are concise and transparent. We then present a linguistic rule induction method from training data based on information coarsening and data clustering. Finally, we apply this linguistic modelling method to some benchmark time series prediction problems, which show that our linguistic modelling and information coarsening methods are potentially powerful tools for linguistic modelling and uncertain reasoning.     

A theory of entropy in Fourier analysis

This thesis deals with a certain set function called entropy and its ápplications to some problems in classical Fourier analysis. For a set S [0, 1/e] the entropy of S is defined by E(S) = inf_{SkIk,Ik intervals} Σ_k | I_k | log(1/|I_k|). We begin by using notions related to entropy in order to investigate the maximal operator M_Ω given by M_Ω(f)(x) = sup_r>0(1/rⁿ) ∫_{|t| ≤r} Ω(t) |f(x + t)| dt, f ε L¹(Rⁿ), where Ω is a positive function, homogeneous of degree 0, and satisfying a certain weak smoothness condition. Then the set function entropy is investigated, and certain of its properties are derived. We then apply these to solve various problems in differentiation theory and the theory of singular integrals, deriving in the process, entropic versions of the theorems of Hardy and Littlewood and Calderón and Zygmund.     

Classical and approximate sampling theorems; studies in the and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for , 1p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to -space, it is shown that the classical sampling theorem for , 1<p<∞ (here p=1 must be excluded), implies the -approximate sampling theorem with convergence in the -norm, provided that f is locally Riemann integrable and belongs to a certain class Λ^p. Basic in the proof is an intricate result on the representation of the integral as the limit of an infinite Riemann sum of |f|^p for a general family of partitions of ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on . These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.     

Compact composition operators on BMOA(<Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

The paper gives some characterization theorems for the compact composition operator on some function spaces over the unit ball B _n in ℂ ⁿ . Especially, it gives a characterization for compact composition operators on BMOA(B _n ), which generalizes a result proved by Bourdon, Cima and Matheson for the case n = 1.     

The Rough Laplacian and Harmonicity of Hopf Vector Fields

Let M be an orientable real hypersurface of a general Kähler manifold . The characteristic vector field ξ of the induced almost contact metric structure (ξ,η, g,ϕ) is also called the Hopf vector field of M. In this paper, we compute the ‘rough’ Laplacian of ξ in terms of the shape operator A and also (as a natural generalization of the contact metric case) in terms of torsion τ = L_ξ g. Then we give some criteria of harmonicity of ξ. Moreover, we consider hypersurfaces M of contact type and give some criteria for M to admit an H-contact structure.Mathematics Subject Classifications (2000): 53C25, 53C20, 53C40, 53D35.     

Characterizations of <Emphasis Type="Italic">r</Emphasis>-Convex Functions

This paper discusses some properties of r-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of r-convexity is given without assuming differentiability. The concept of strict r-convexity is introduced. For a twice continuously differentiable function f, it is shown that the strict r-convexity of f is equivalent to a certain condition on ∇ ² f. Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.     

Generalized p-Center problems: Complexity results and approximation algorithms

In an earlier paper, two alternative p-Center problems, where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the ‘Set’ p-Center problems. We establish that the triangle inequality (Δ-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two Δ-inequality Set p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless P = NP. For the special case ‘alternative’ p-Center problems, which we refer to as the ‘Pair’ p-Center problems, we extend the previous results in several ways. For example, the results mentioned above for the Set p-Center problems also apply to the Pair p-Center problems. Furthermore, we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the Pair p-Center problems.     

Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols

The asymptotics for determinants of Toeplitz and Wiener-Hopf operators with piecewise continuous symbols are obtained in this paper. If W_α(σ) is the Wiener-Hopf operator defined on L₂(0, α) with piecewise continuous symbol σ having a finite number of discontinuities at ξ_r, then under appropriate conditions it is shown that det W_α(σ) ˜ G(σ)^α α^Σλ_r2K(σ), where

Full-size image

is a completely determined constant. An analogous result is obtained for Toeplitz operators. The main point of the paper is to obtain a result in the Wiener-Hopf case since the Toeplitz case had been treated earlier. In the Toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if, for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known. Fortunately in the Toeplitz case such asymptotics were known. The difficulty in the Wiener-Hopf case is that there was not a single singular case where the determinant was explicitly known. This problem was overcome by using the fact that Wiener-Hopf determinants when discretized become Toeplitz determinants whose entries depend on the size of the matrix. No theorem on Toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results.     

An algebraical system for polynomial representation of k-valued logical functions

An algebraical system < A_k;+; >, in which ‘almost each’ k-valued logical function (k-composite integer) had representation in the form of polynomial in modulo k, is presented.     

A correction to the definition of local center

It is known that in order to solve the minimax facility location problem on a graph with a finite set of demand points, only a finite set of possible location points, called ‘local centers’ must be considered.It has been shown that the continuous m-center problem on a graph can be solved by using a series of set covering problems in which each local center covers the demand points at a distance not greater than a corresponding number called ‘the range’ of the local center.However, all points which are at the same distance from more than two demand points, and from which there is no direction where all these distances are decreasing, must also be considered as local centers. This paper proves that, in some special cases, it is not sufficient to consider only the points where this occurs with respect to pairs of demand points. The definition of local center is corrected and the corresponding results and algorithms are revised.