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

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 ,

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.     

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.     

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.     

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.     

Homogenization of Non-Linear Variational Problems with Thin Low-Conducting Layers

This paper deals with the homogenization of a sequence of non-linear conductivity energies in a bounded open set The energy density is of the same order as where is periodic, u is a vector-valued function in and The conductivity is equal to 1 in the "hard" phases composed by two by two disjoint-closure periodic sets while tends uniformly to 0 in the "soft" phases composed by periodic thin layers which separate the hard phases. We prove that the limit energy, according to γ-convergence, is a multi-phase functional equal to the sum of the homogenized energies (of order 1) induced by the hard phases plus an interaction energy (of order 0) due to the soft phases. The number of limit phases is less than or equal to N and is obtained by evaluating the γ-limit of the rescaled energy of density in the torus. Therefore, the homogenization result is achieved by a double γ-convergence procedure since the cell problem depends on ε.     

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.     

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.     

Convolution Products in Naturally Ordered Semigroups

In this paper new equalities between two different convolution products in cancellative naturally ordered semigroups (but not in groups) are given. We also give several applications in particular cases and      

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].     

Staircase Connected Sets

A compact set is staircase connected if every two points can be connected by a polygonal path with sides parallel to the coordinate axes, which is both x-monotone and y-monotone. denotes the smallest number of edges of such a path. is an integer-valued metric on S. We investigate this metric and introduce stars and kernels. Our main result is that the r-th kernel is nonempty, compact and staircase connected provided .     

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.     

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.     

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.     

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.     

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      

On the Reproducing Formula of Calderon

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

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).