On Sets of Unbounded Divergence at Each Point of Multiple Fourier Series of a Function Equal Zero on a Closed Set

The problem investigated is to characterize sets E, the sets of unbounded divergence (at each point) of single and multiple Fourier series under condition of convergence of these series to zero at each point of the complement of E.For any nonempty open set B T ^N = [0, 2] ^N , N 1, a Lebesgue integrable function f ₀ is constructed which equals zero on the set U = T ^N \ B whose multiple trigonometric Fourier series diverges unboundedly (in the case of summation over squares) at each point of the set

Fast solution methods for fredholm integral equations of the second kind

Summary The main purpose of this paper is to describe a fast solution method for one-dimensional Fredholm integral equations of the second kind with a smooth kernel and a non-smooth right-hand side function. Let the integral equation be defined on the interval [–1, 1]. We discretize by a Nyström method with nodes {cos(j/N)} _j ^=0/N . This yields a linear system of algebraic equations with an (N+1)×(N+1) matrixA. GenerallyN has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation. We show by Fourier analysis thatA can be approximated well by , a low-rank modification of the identity matrix. ReplacingA by in the linear system of algebraic equations yields a new linear system of equations, whose elements, and whose solution , can be computed inO (N logN) arithmetic operations. If the kernel has two more derivatives than the right-hand side function, then is shown to converge optimally to the solution of the integral equation asN increases.We also consider iterative solution of the linear system of algebraic equations. The iterative schemes use bothA andÃ. They yield the solution inO (N ²) arithmetic operations under mild restrictions on the kernel and the right-hand side function.Finally, we discuss discretization by the Chebyshev-Galerkin method. The techniques developed for the Nyström method carry over to this discretization method, and we develop solution schemes that are faster than those previously presented in the literature. The schemes presented carry over in a straightforward manner to Fredholm integral equations of the second kind defined on a hypercube.     

An Entire Holomorphic Function Associated to an Entire Harmonic Function

Let h be a harmonic function on ^N. Then there exists a holomorphic function f on such that f(t)=h(t, 0, …, 0) for all real t. Precise inequalities relating the growth rate of f to that of h are proved. These results are applied to deduce uniqueness theorems for harmonic functions of sufficiently slow growth that vanish at certain lattice points. Another application concerns the rate at which a harmonic function of finite order can decay along a ray.     

Davenport's Theorem in the Theory of Irregularities of Point Distribution

We study distributions of N points in the unit square U ² with minimal order of L ₂-discrepancy ₂[ ] < C(log N)^1/2, where the constant C is independent of N. We present an approach that uses Walsh functions and can be generalized to higher dimensions. Bibliography: 19 titles.     

On Subfields of the Hermitian Function Field

The Hermitian function field H= K(x,y) is defined by the equationy ^q+ y=x ^q+1(q being a powerof the characteristic of K). OverK= q ² it is a maximalfunction field; i.e. the numberN(H)of q²-rationalplaces attains the Hasse--Weil upper boundN(H)=q ²+1+2g(H)·q.All subfields K EHare also maximal.In this paper we construct a large number of nonrational subfields EH, by considering the fixed fieldsH under certaingroups type="Italic">g0 that occur as the genus of some maximal function field over q ².     

Nonconvergence results for the application of least-squares estimation to Ill-posed problems

One standard approach to solvingf(x)=b is the minimization of f(x)–b² overx in , where corresponds to a parametric representation providing sufficiently good approximation to the true solutionx*. Call the minimizerx=A( ). Take = _N for a sequence { _N } of subspaces becoming dense, and so determine an approximating sequences {x _N A ( _N )}. It is shown, withf linear and one-to-one, that one need not havex _Nx* iff ^–1 is not continuous.This work was supported by the US Army Research Office under Grant No. DAAG-29-77-G-0061. The author is indebted to the late W. C. Chewning for suggesting the topic in connection with computing optimal boundary controls for the heat equation (Ref. 2).     

Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric

We consider supercritical two-dimensional Bernoulli percolation. Conditionally on the event that the open cluster C containing the origin is finite, we prove that: the laws of C/N satisfy a large deviations principle with respect to the Hausdorff metric; let f(N) be a function from to such that f(N)/ln N+ and f(N)/N0 as N goes to the laws of {x ² : d(x, C)f(N)}/N satisfy a large deviations principle with respect to the L ¹ metric associated to the planer Lebesgue measure. We link the second large deviations principle with the Wulff construction.     

Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids

We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) φ(r)=r^γ for γ>0, γ2 or φ(r)=r^γ ln r for γ2 ₊. For each positive integer N, let h=N⁻¹ and let {x_i: i =1, 2, …, (N+1)^d} be the set of vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0, 1]^d. Given f: [0, 1]^d→ , let s_h be its unique RBF interpolant at the grid vertices: s_h(x_i)=f(x_i), i=1, 2, …, (N+1)^d. For h→0, we show that the uniform norm of the error f−s_h on a compact subset K of the interior of [0, 1]^d enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid h ^d, provided that f is a data function whose partial derivatives in the interior of [0, 1]^d up to a certain order can be extended to Lipschitz functions on [0, 1]^d.     

L 2 harmonic forms and stability of hypersurfaces with constant mean curvature

We prove that a complete noncompact oriented strongly stable hypersurfaceM ⁿ with cmc (constant mean curvature)H in a complete oriented manifoldN ⁿ⁺¹ with bi-Ricci curvature, satisfying alongM, admits no nontrivialL ² harmonic 1-forms. This implies ifM ⁿ (2n4) is a complete noncompact strongly stable hypersurface in hyperbolic spaceH ⁿ⁺¹(–1) with cmc , there exist no nontrivialL ² harmonic 1-forms onM. We also classify complete oriented strongly stable surfaces with cmcH in a complete oriented manifoldN ³ with scalar curvature satisfying .     

Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scaleN

In this paper we show how wavelets originating from multiresolution analysis of scaleN give rise to certain representations of the Cuntz algebrasO _N , and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space by (S _i ) (z)=m _i (z)(z ^N ). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over . This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations ofO _N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.Work supproted in part by the U.S. National Science Foundation and the Norwegian Research Council.     

Law of the iterated logarithm for perturbed empirical distribution functions evaluated at a random point for nonstationary random variables

We consider perturbed empirical distribution functions , where {Ginn, n1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and {X _{i, i}1} is a non-stationary sequence of absolutely regular random variables. We derive the almost sure representation and the law of the iterated logarithm for the statistic whereU _n is aU-statistic based onX ₁,...,X _n . The results obtained extend or generalize the results of Nadaraya,⁽⁷⁾ Winter,⁽¹⁶⁾ Puri and Ralescu,^(9,10) Oodaira and Yoshihara,⁽⁸⁾ and Yoshihara,⁽¹⁹⁾ among others.Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

Extreme norms on ℝ n

LetN ¹,N be thel ¹ andl norms on ⁿ . We denote by the set of all normsN on ⁿ such thatN NN ¹. The aim of the paper is to present a characterization of the extreme points of .This paper has been written while the author was a research fellow of the Alexander von Humboldt-Stiftung at the Mathematisches Institut der Eberhardt-Karls-Universität, Tübingen.     

A nonexistence result for abelian menon difference sets using perfect binary arrays

A Menon difference set has the parameters (4N ² ,2N ² -N, N ² -N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group contains a Menon difference set, wherep is an odd prime, |K|=p , andp ^j–1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group , where exp(H)=2 or 4 and |K|=3, if and only ifK is cyclic.This work is partially supported by NSA grant # MDA 904-92-H-3057 and by NSF grant # NCR-9200265. The author thanks the Mathematics Department, Royal Holloway College, University of London for its hospitality during the time of this researchThis work is partially supported by NSA grant # MDA 904-92-H-3067     

On the set visited once by a random walk

Summary In this paper we prove the following statement. Given a random walk ,n=1, 2, ... where ₁, ₂ ... are i.i.d. random variables, let (n) denote the number of points visited exactly once by this random walk up to timen. We show that there exists some constantC, 0 <C < , such that with probability 1. The proof applies some arguments analogous to the techniques of the large deviation theory.Research supported by the Hungarian National Foundation for Scientific Research, Grant N^o # 819/1     

Geometry of \mathbb{Z}^d and the Central Limit Theorem for Weakly Dependent Random Fields

We study the asymptotic distribution of where A is a subset of , A _N = A[–N, N] ^d , v(A) = lim _N card(A _N) (2N+1) ^–d (0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S _N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if has a limit F(n; A) as N for each . We also study the class of sets A that satisfy this condition.     

Topology of the Real Part of the Hyperelliptic Jacobian Associated with the Periodic Toda Lattice

We study the topology of the isospectral real manifold of the periodic Toda lattice consisting of 2 ^N–1 different systems. The solutions of these systems contain blow-ups, and the set of these singular points defines a divisor of the manifold. With the divisor added, the manifold is compactified as the real part of the (N–1)-dimensional Jacobi variety associated with a hyperelliptic Riemann surface of genus g=N–1. We also study the real structure of the divisor and provide conjectures on the topology of the affine part of the real Jacobian and on the gluing rule over the divisor to compactify the manifold based on the sign representation of the Weyl group of .     

Riemannian Submersions and Riemannian Manifolds with Einstein--Weyl Structures

The main results of this paper are as follows. (a) Let : M N be a non-trivial Riemannian submersion with totally geodesic fibers of dimension 1 over an Einstein manifold N. If M is compact and admits a standard Einstein--Weyl structure with constant Einstein--Weyl function, then N admits a Kähler structure andM a Sasakian structure. (b) Let be a Riemannian submersion with totally geodesic fibers and N an Einstein manifold of positive scalar curvature . If M admits a standard Sasakian structure, then M admits an Einstein--Weyl structure with constant Einstein--Weyl function.     

Optimal Domains and Integral Representations of Convolution Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">G</Emphasis>)

Given , a compact abelian group G and a function , we identify the maximal (i.e. optimal) domain of the convolution operator (as an operator from L^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L^p(G) is embedded and to which has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain. Several characterizations of precisely when this is the case are presented.     

Containing and shrinking ellipsoids in the path-following algorithm

We describe a new potential function and a sequence of ellipsoids in the path-following algorithm for convex quadratic programming. Each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors. Furthermore, the volumes of the ellipsoids shrink at the ratio , in comparison to 2^–(1) in Karmarkar's algorithm and 2^–(1/n) in the ellipsoid method. We also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming.Research supported by The U.S. Army Research Office through The Mathematical Sciences Institute of Cornell University when the author was visiting at Cornell.Research supported in part by National Science Foundation Grant ECS-8602534 and Office of Naval Research Contract N00014-87-K-0212.     

Irrepresentability of short semilattices by euclidean subspaces

A semilatticeS isrepresentable by subspaces of R ^k if, to eachx S we can assign a subspace so thatx y=z inS if and only if . Every height-2 semilattice is representable inR ². We show that for everyk there is a height-3 semilattice which is not representable by subspaces ofR ^k.Presented by J. Berman.Research supported in part by the National Science Foundation.Research supported in part by the Office of Naval Research.