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.     

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.     

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.     

Spherical Perturbations of Schrödinger Equations

We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights and on , for which the a priori inequality holds with constant independent of . Here is the Laplace-Beltrami operator on the sphere . Due to the relation between and the tangential component of the gradient, , we obtain some "Morawetz-type" estimates for on . As a consequence we establish some new estimates for the free Schr?dinger propagator , which may be viewed as certain refinements of the -(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem for certain time dependent first order spherical perturbations of the dimensional Schr?dinger equation.     

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.     

Hardy Spaces on the Plane and Double Fourier Transforms

We provide a direct computational proof of the known inclusion where is the product Hardy space defined for example by R. Fefferman and is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in and respectively. In particular, we obtain a broad class of multipliers on and respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on and respectively.     

Density of the Set of Generators of Wavelet Systems

Given a function ψ in the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions In this paper we prove that the set of functions generating affine systems that are a Riesz basis of ${\cal L}^2({\Bbb R}^d)$ is dense in We also prove that a stronger result is true for affine systems that are a frame of In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems.     

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If and , and is locally integrable, then distributionally if and only if there exists k such that , for each a > 0, and similarly in the case when is a general distribution. Here means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by . We also show that under some extra conditions, as if the sequence belongs to the space for some and the tails satisfy the estimate ,\ as , the asymmetric partial sums\ converge to . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.     

Regular Semigroups with Inverse Transversals

Let C be a semiband with an inverse transversal . In [7], G.T. Song and F.L. Zhu construct a fundamental regular semigroup with an inverse transversal . is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation-although not the usual composition-is defined by means of composition. Any full regular subsemigroup T of is a fundamental regular semigroup with inverse transversal . Moreover, any regular semigroup S with an inverse transversal is proved to be an idempotent-separating coextension of a full regular subsemigroup T of some . By means of a full regular subsemigroup T of some and by means of an inverse semigroup K satisfying some conditions, in this paper, we construct a regular semigroup with inverse transversal such that is isomorphic to K and to T. Furthermore, it is proved that if S is a regular semigroup with an inverse transversal then S can be constructed from the corresponding T and from in this way.     

Duality and Biorthogonality for Weyl-Heisenberg Frames

Let and let In this paper we investigate the relation between the frame operator and the matrix whose entries are given by for Here , for any We show that is bounded as a mapping of into if and only if is bounded as a mapping of into Also we show that if and only if where denotes the identity operator of and respectively, and Next, when generates a frame, we have that has an upper frame bound, and the minimal dual function can be computed as The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a generating a frame are inherited by In particular, we show that when generates a frame Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.     

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].     

Local Growth Envelopes of Triebel-Lizorkin Spaces of Generalized Smoothness

The concept of local growth envelope of the quasi-normed function space is applied to the Triebel-Lizorkin spaces of generalized smoothness In order to achieve this, a standardization result for these and corresponding Besov spaces is derived.     

Nonlinear Approximation by Trigonometric Sums

We investigate the -error of approximation to a function by a linear combination of exponentials on where the frequencies are allowed to depend on We bound this error in terms of the smoothness and other properties of and show that our bounds are best possible in the sense of approximation of certain classes of functions.     

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.     

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

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., .     

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      

Approximation and Spanning in the Hardy Space,by Affine Systems

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer , provided only that and satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each ,

Old and New Morrey Spaces with Heat Kernel Bounds

Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space of all locally integrable complex-valued functions f on such that for every open Euclidean ball B ⊂ with radius r_B there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying