Numerical Integration over Spheres of Arbitrary Dimension

In this paper we study the worst-case error (of numerical integration) on the unit sphere for all functions in the unit ball of the Sobolev space where More precisely, we consider infinite sequences of m(n)-point numerical integration rules where: (i) is exact for all spherical polynomials of degree and (ii) has positive weights or, alternatively to (ii), the sequence satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) in has the upper bound where the constant c depends on s and d (and possibly the sequence This extends the recent results for the sphere by K. Hesse and I.H. Sloan to spheres of arbitrary dimension by using an alternative representation of the worst-case error. If the sequence of numerical integration rules satisfies an order-optimal rate of convergence is achieved.     

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

Discretization and Transference of Bisublinear Maximal Operators

We develop a general condition for automatically discretizing strong type bisublinear maximal estimates that arise in the context of the real line. In particular, this method applies directly to Michael Lacey’s strong type boundedness results for the bisublinear maximal Hilbert transform and for the bisublinear Hardy-Littlewood maximal operator, furnishing the counterpart of each of these two results (without changes to the range of exponents) for the sequence spaces We then take up some transference applications of discretized maximal bisublinear operators to maximal estimates and almost everywhere convergence in Lebesgue spaces of abstract measures. We also broaden the scope of such applications, which are based on transference from by developing general methods for transplanting bisublinear maximal estimates from arbitrary locally compact abelian groups.     

Fourier Inequalities and Moment Subspaces in Weighted Lebesgue Spaces

For define where Pointwise estimates and weighted inequalities describing the local Lipschitz continuity of are established. Sufficient conditions are found for the boundedness of from into and a spherical restriction property is proved. A study of the moment subspaces of is next developed in the one-variable case, for locally integrable, a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in Characterizations are also given for each class. Applications related to the approximation and decomposition of are discussed.     

Improved Approximation Algorithms for Geometric Set Cover

Given a collection S of subsets of some set and the set cover problem is to find the smallest subcollection that covers that is, where denotes We assume of course that S covers While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset and nondecreasing function f(·), there is a decomposition of the complement into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|)) can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in and for guarding an x-monotone polygonal chain.     

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

In this paper we develop a robust uncertainty principle for finite signals in which states that, for nearly all choices such that

The Cayley Transform of the Generator of a Bounded C<Subscript>0</Subscript>-Semigroup

Let A be the generator of a uniformly bounded C₀-semigroup on the Banach space X. We present sufficient conditions on the resolvent , under which the Cayley transform is a power-bounded operator, i.e., .     

Affine pseudo-planes with torus actions

An affine pseudo-plane X is a smooth affine surface defined over which is endowed with an -fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic -action on X and X is an -surface, we shall show that the universal covering is isomorphic to an affine hypersurface in the affine 3-space and X is the quotient of by the cyclic group via the action where and It is also shown that a -homology plane X with and a nontrivial -action is an affine pseudo-plane. The automorphism group is determined in the last section.     

Frame Decomposition of Decomposition Spaces

A new construction of tight frames for with flexible time-frequency localization is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on a class of so-called decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space, anisotropic Besov spaces, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type tight frames are constructed on      

A Two-Dimensional Kinetic Triangulation with Near-Quadratic Topological Changes

A triangulation of a set S of points in the plane is a subdivision of the convex hull of S into triangles whose vertices are points of S. Given a set S of n points in each moving independently, we wish to maintain a triangulation of S. The triangulation needs to be updated periodically as the points in S move, so the goal is to maintain a triangulation with a small number of topological events, each being the insertion or deletion of an edge. We propose a kinetic data structure (KDS) that processes topological events with high probability if the trajectories of input points are algebraic curves of fixed degree. Each topological event can be processed in time. This is the first known KDS for maintaining a triangulation that processes a near-quadratic number of topological events, and almost matches the lower bound [1]. The number of topological events can be reduced to if only k of the points are moving.     

Isometry-Invariant Valuations on Hyperbolic Space

Hyperbolic area is characterized as the unique continuous isometry-invariant simple valuation on convex polygons in We then show that continuous isometry-invariant simple valuations on polytopes in for are determined uniquely by their values at ideal simplices. The proofs exploit a connection between valuation theory in hyperbolic space and an analogous theory on the Euclidean sphere. These results lead to characterizations of continuous isometry-invariant valuations on convex polytopes and convex bodies in the hyperbolic plane a partial characterization in and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas, containment theorems, and isoperimetric and Bonnesen-type inequalities.     

Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.     

A Numerical Method for Constructing the Hyperbolic Structure of Complex Henon Mappings

For complex parameters a,c, we consider the Henon mapping given by and its Julia set, J. In this paper we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J, which is preserved by DH, and a continuous norm in which uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is {hyperbolic} on J. We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.     

Generalized Bessel Functions for p-Radial Functions

Suppose that and p > 0. In this paper we study the generalized Bessel functions for the surface , introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation to relate them to the classical symmetric functions. These generalized Bessel functions are symmetric with respect to the action of the hyperoctahedral group W_d, which is the symmetry group of the unit sphere. By means of this symmetry under W_d, we further express these generalized Bessel functions in terms of Bessel functions for certain finite reflection groups. For the case in which p = 2, our representations lead to known relations for the classical Bessel functions of order (d - 2)/2. For the case in which p = 1, the generalized Bessel functions have been studied by Berens and Xu in the analysis of summability problems for 1-radial functions, and we show how their results may be framed within our more general context.     

On the Reproducing Formula of Calderon

Let and Under certain conditions on we shall prove that converges nontangentially to at for      

Inequalities for irregular Gabor frames

In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if is a Gabor frame for with frame bounds A and B, then the following two inequalities hold: and . In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form , where Δ _k and Λ _k are arbitrary sequences of points in and , 1 ≤ k ≤ r. Corresponding author for second author Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China     

Row Space Cardinalities

Let be the set of all Boolean matrices. Let R(A) denote the row space of , let , and let . By extensive computation we found that

Pairs of Explicitly Given Dual Gabor Frames in L2(ℝd)

Given certain compactly supported functions g ≥ L²(ℝ^d) whose ℤ^d-translates form a partition of unity, and real invertible d × d matrices B,C for which ||C^T B|| is sufficiently small, we prove that the Gabor system forms a frame, with a (noncanonical) dual Gabor frame generated by an explicitly given finite linear combination of shifts of g. For functions g of the above type and arbitrary real invertible d × d matrices B,C this result leads to a construction of a multi-Gabor frame , where all the generators g_k are dilated and translated versions of g. Again, the dual generators have a similar form, and are given explicitly. Our concrete examples concern box splines.     

Hypoelliptic Dunkl Convolution Equations in the Space of Distributions on {\Bbb R}^d

In this article we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.