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.     

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems

Let be a nontrivial probability measure on the unit circle the density of its absolutely continuous part, its Verblunsky coefficients, and its monic orthogonal polynomials. In this paper we compute the coefficients of in terms of the . If the function is in , we do the same for its Fourier coefficients. As an application we prove that if and if is a polynomial, then with and S the left-shift operator on sequences we have

Line Transversals to Translates of Unit Discs

Let be a family of convex figures in the plane. We say that has property T if there exists a line intersecting every member of . Also, the family has property T(k) if every k-membered subfamily of has property T. Let B be the unit disc centered at the origin. In this paper we prove that if a finite family of translates of B has property T(4) then the family , where , has property T. We also give some results concerning families of translates of the unit disc which has either property T(3) or property T(5).     

Orthogonal Algebraic Polynomial Schauder Bases of Optimal Degree

For any fixed we construct an orthonormal Schauder basis for C[-1,1] consisting of algebraic polynomials with The orthogonality is with respect to the Chebyshev weight.     

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.     

Direct Sum Decomposition of L2(Rn 1) into Subspaces Invariant under Fourier Transformation

Denote by the real-linear span of , where Under the concept of left-monogeneity defined through the generalized Cauchy-Riemann operator we obtain the direct sum decomposition of

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.     

Kernel and Trace Operators for Extensions of Brandt Semigroups

Let S be an (ideal) extension of a Brandt semigroup S₀ by a Brandt semigroup S₁ and let denote the congruence lattice of S. For denote by and the least and the greatest congruences on S with the same kernel as respectively, and let and have the analogous meaning relative to trace. We establish necessary and sufficient conditions on S in order that one or more of the operators

Hardy Spaces for Laguerre Expansions

Let be the standard Laguerre functions of type a. We denote . Let and be the semigroups associated with the orthonormal systems and . We say that a function f belongs to the Hardy space associated with one of the semigroups if the corresponding maximal function belongs to . We prove special atomic decompositions of the elements of the Hardy spaces.     

Standard monomial bases, Moduli spaces of vector bundles, and Invariant theory

Consider the diagonal action of on the affine space where an algebraically closed field of characteristic We construct a "standard monomial" basis for the ring of invariants As a consequence, we deduce that is Cohen-Macaulay. As the first application, we present the first and second fundamental theorems for -actions. As the second application, assuming that the characteristic of K is we give a characteristic-free proof of the Cohen-Macaulayness of the moduli space of equivalence classes of semi-stable, rank 2, degree 0 vector bundles on a smooth projective curve of genus > 2. As the third application, we describe a K-basis for the ring of invariants for the adjoint action of on m copies of in terms of traces.     

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.     

Different Digitisations of Displaced Discs

The digitisation of a real disc having radius and centre consists of all integer points inside , i.e., In this paper we show that there are

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Given an invertible matrix B and a finite or countable subset of , we consider the collection generating the closed subspace of . If that collection forms a frame for , one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively. The main distinction between them is that a SG-dual of type I is required to be contained in the space generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and are known to play an important role in the theory of shift-invariant spaces.     

Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations

This paper studies intersection theory on the compactified moduli space of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of Based on our earlier work [25], we give a precise formula for the intersection cohomology pairings and provide a method to calculate pairings on The case when n = 2 is discussed in detail. Finally Witten's integral is considered for this singular case.     

On the Bernstein Constants of Polynomial Approximation

Assume is not an integer. In papers published in 1913 and 1938, S.~N.~Bernstein established the limit

The Entropy in Learning Theory. Error Estimates

We continue the investigation of some problems in learning theory in the setting formulated by F. Cucker and S. Smale. The goal is to find an estimator on the base of given data that approximates well the regression function of an unknown Borel probability measure defined on We assume that belongs to a function class It is known from previous works that the behavior of the entropy numbers of in the uniform norm plays an important role in the above problem. The standard way of measuring the error between a target function and an estimator is to use the norm ( is the marginal probability measure on X generated by ). This method has been used in previous papers. We continue to use this method in this paper. The use of the norm in measuring the error has motivated us to study the case when we make an assumption on the entropy numbers of in the norm. This is the main new ingredient of thispaper. We construct good estimators in different settings: (1) we know both and ; (2) we know but we do not know and (3) we only know that is from a known collection of classes but we do not know An estimator from the third setting is called a universal estimator.     

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

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.     

Invariance Theorems in Approximation Theory and their Applications

Let B be a closed linear subspace of a Banach space F and let be a group of continuous linear operators , where G is a compact topological group. We prove that if is invariant under , then under some conditions on f, F, B, and G, there exists an element of best approximation to f that has the same property. As applications, we compute the bivariate Bernstein constant for polynomial approximation of and solve a Braess problem on the exponential order of decay of the error of polynomial approximation of . Other examples and applications are discussed as well.