Rep-tiling Euclidean space

Summary Arep-tiling is a self replicating, lattice tiling ofR ⁿ .Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, andself-replicating means that there is a non-singular linear mapø: R ⁿ Rⁿ such that, for eachT , the imageø(T) is, in turn, tiled by . This topic has recently come under investigation, not only because of its recreational appeal, but because of its application to the theory of wavelets and to computer addressing. The paper presents an exposition of some recent results on rep-tiling, including a construction of essentially all rep-tilings of Euclidean space. The construction is based on radix representation of points of a lattice. One particular radix representation, called thegeneralized balanced ternary, is singled out as an example because of its relevance to the field of computer vision.     

Regarding thep-norms of radial basis interpolation matrices

A radial basis function approximation has the form where:R ^d R is some given (usually radially symmetric) function, (y _j ) ₁ ⁿ are real coefficients, and the centers (x _j ) ₁ ⁿ are points inR ^d . For a wide class of functions , it is known that the interpolation matrixA=((x _j –x _k )) _j,k=1 ⁿ is invertible. Further, several recent papers have provided upper bounds on ||A ^–1||₂, where the points (x _j ) ₁ ⁿ satisfy the condition ||x _j –x _k ||₂,jk, for some positive constant . In this paper we calculate similar upper bounds on ||A ^–1||₂ forp1 which apply when decays sufficiently quickly andA is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrixA _n = ((j–k)) _j,k=1 ⁿ when (x)=(x ²+c ²)^1/2, the Hardy multiquadric. In particular, we show that sup _n ||A _n ^–1 || is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix enjoys the remarkable property that ||E ^–1|| _p = ||E ^–1||₂ for everyp1 when is a Gaussian. Indeed, we also show that this property persists for any function which is a tensor product of even, absolutely integrable Pólya frequency functions.Communicated by Charles Micchelli.     

Bases and closures under infinite sums

Motivated by work of Diestel and Kühn on the cycle spaces of infinite graphs we study the ramifications of allowing infinite sums in a module R^M. We show that every generating set in this setup contains a basis if the ground set M is countable, but not necessarily otherwise. Given a family N⊆R^M, we determine when the infinite-sum span N of N is closed under infinite sums, i.e. when N=N. We prove that this is the case if R is a field or a finite ring and each element of M lies in the support of only finitely many elements of N. This is, in a sense, best possible. We finally relate closures under infinite sums to topological closures in the product space R^M.     

Conditionally positive functions andp-norm distance matrices

In [4], deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated byp-norms. In particular, we show that, for anyp(1, 2), and for distinct pointsx ¹,,x ^{n d} , wheren andd may be any positive integers, with the proviso thatn2, the matrixA ^n×n defined by

Polar decompositions and related classes of operators in spaces ∏κ

Polar decompositions with respect to an indefinite inner product are studied for bounded linear operators acting on a space. Criteria are given for existence of various forms of the polar decompositions, under the conditions that the range of a given operatorX is closed and that zero is not an irregular critical point of the selfadjoint operatorX ^[*]X. Both real and complex spaces are considered. Relevant classes of operators having a selfadjoint (in the sense of the indefinite inner product) square root, or a selfadjoint logarithm, are characterized.The work of this author was partially supported by INdAM-GNCS and MURSTThe work of this author was partially supported by NSF grant DMS-9988579.     

Invariance Principles for Energy Functionals on Spheres

We study a generalised version of the g-energy functionals defined by Damelin and Grabner. We comment on invariance principles for finite energies and use these principles to obtain expansions of these latter energies in terms of cap discrepancies for a subclass of g. This allows for discrepancy estimates knowing bounds on the energy and vice versa. We are, in particular, able to carefully analyse the case when g gives a Riesz kernel g^R_s when 0<sd or a logarithmic kernel g^L₀ in the limits when 0⁺.The author is supported by the START project Y96-MAT of the Austrian Science Fund.     

Asymptotic expansion formula for Bernstein polynomials defined on a simplex

In this paper we give a complete expansion formula for Bernstein polynomials defined on ans-dimensional simplex. This expansion for a smooth functionf represents the Bernstein polynomialB _n (f) as a combination of derivatives off plus an error term of orderO(n^–s ).Communicated by Wolfgang Dahmen.     

Divisors of Zero in the Lipschitz Semigroup

The Lipschitz semigroup is generated by all (invertible and noninvertible) Clifford vectors. We show that all solutions of the equation xy = 0 (where x, y are non-zero elements of the Lipschitz semigroup) are of the form x = av₀, y = v₀b where v₀ is an isotropic vector (i.e., v₀² = 0). This problem turns out to be useful in the construction of multisoliton solutions of integrable systems of nonlinear partial differential equtions.     

Approximation in L p (R d ) from a space spanned by the scattered shifts of a radial basis function

A new multivariate approximation scheme on R ^d using scattered translates of the “shifted” surface spline function is developed. The scheme is shown to provide spectral L _p -approximation orders with 1 ≤ p ≤ ∞, i.e., approximation orders that depend on the smoothness of the approximands. In addition, it applies to noisy data as well as noiseless data. A numerical example is presented with a comparison between the new scheme and the surface spline interpolation method.     

Random k-dimensional orders: Width and number of linear extensions

We consider the width W _k (n) and number L _k (n) of linear extensions of a random k-dimensional order P _k (n). We show that, for each fixed k, almost surely W _k (n) lies between (k/2–C)n _1–1/k and 4kn ^1-1/k , for some constant C, and L _k (n) lies between (e ^-2 n ^1-1/k ) ⁿ and (2kn ^1-1/k ) ⁿ . The bounds given also apply to the expectations of the corresponding random variables. We also show that W _k (n) and log L _k (n) are sharply concentrated about their means.     

Quadratische Identitäten bei Polygonen

Summary In a letter to Christian Goldbach (1690–1764) Leonhard Euler (1707–1783) communicates the following theorem on quadrilaterals: IfABCD is a quadrilateral andM, N are the mid-points of the diagonals, then |AB|²+|BC|²+|CD|²+|DA|²=|AC|²+|BD|²+4|MN|². This theorem, a generalization of a classical theorem of Apollonius of Perge (262-190 B.C.) on parallelograms, was rediscovered recently by A. R. Amir-Moez and J. D. Hamilton (1976). A. J. Douglas (1981) carried the generalization a stage further and proved a theorem on polygons in a Euclidean space, which have an even number of points. On the basis of the Fourier analysis of polygons this paper establishes a wide class of quadratic identities, whose geometrical interpretation leads to polygon-theorems of the above type.

Meiner Mutter zum 85. Geburtstag gewidmet     

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ) such thatu ^–1 is operator monotone. We then look for another functionv such thatv(u ^–1) is operator monotone, namely,u(A)u(B) impliesv(A)v(B) for self-adjoint operatorsA andB.     

Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

Analysis of Alternative Digit Sets for Nonadjacent Representations

It is known that every positive integer n can be represented as a finite sum of the form ∑_ia_i2ⁱ, where a_i ∈ {0, 1,−1} and no two consecutive a_i’s are non-zero (“nonadjacent form”, NAF). Recently, Muir and Stinson [14, 15] investigated other digit sets of the form {0, 1, x}, such that each integer has a nonadjacent representation (such a number x is called admissible). The present paper continues this line of research. The topics covered include transducers that translate the standard binary representation into such a NAF and a careful topological study of the (exceptional) set (which is of fractal nature) of those numbers where no finite look-ahead is sufficient to construct the NAF from left-to-right, counting the number of digits 1 (resp. x) in a (random) representation, and the non-optimality of the representations if x is different from 3 or −1. This paper was written while the first author was a visitor at the John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg. He thanks the centre for its hospitality. He was also supported by the grant S8307-MAT of the Austrian Science Fund. This author is supported by the grant NRF 2053748 of the South African National Research Foundation. The research of this author was done while he was with the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg.     

Using the refinement equation for the construction of pre-wavelets III: Elliptic splines

The purpose of this paper is to provide multiresolution analysis, stationary subdivision and pre-wavelet decomposition onL ²(R ^d ) based on a general class of functions which includes polyharmonic B-splines.The work of this author has been partially supported by a DARPA grant.The work of this author has been partially supported by Fondo Nacional de Ciencia y Technologia under Grant 880/89.     

S-convex,monotone, and positive interpolation with rational bicubic splines of C2-continuity

In this paper we deal with shape preserving interpolation of data sets given on rectangular grids. The aim is to show that there exist spline interpolants of the continuity classC ² which areS-convex, monotone, or positive if the data sets have these properties. This is done by using particular rational bicubic splines defined on the grids introduced by the data. Interpolants of the desired type can be constructed by a simple search procedure.     

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

The fundamentality of sets of ridge functions

Summary We investigate the fundamentality of the set of all continuous ridge functions in the spaceC( ⁿ ) as well as inC(X) for a general Banach space,X. Both positive and negative results are obtained. Necessary and sufficient conditions for the fundamentality are given for certain sets of ridge functions inC( ⁿ ).     

Square of ω-hyponormal operators

In this paper, we show that ifT is a -hyponormal operator, thenT ² is also -hyponormal.     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.