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

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].     

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.     

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.     

Localized Polynomial Frames on the Ball

Almost exponentially localized polynomial kernels are constructed on the unit ball in with weights , by smoothing out the coefficients of the corresponding orthogonal projectors. These kernels are utilized to the design of cubature formulas on with respect to and to the construction of polynomial tight frames in (called needlets) whose elements have nearly exponential localization.     

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.     

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.     

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 ,

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.     

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.     

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.     

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.     

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.     

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

A Lattice of Quasivarieties of Normal Cryptogroups

A normal cryptogroup S is a completely regular semigroup in which is a congruence and is a normal band. We represent S as a strong semilattice of completely simple semigroups, and may set For each we set and represent by means of an h-quintuple These parameters are used to characterize certain quasivarieties of normal cryptogroups. Specifically, we construct the lattice of quasivarieties generated by the (quasi)varieties and This is the lattice generated by the lattice of quasivarieties of normal bands, groups and completely simple semigroups. We also determine the B-relation on the lattice of all quasivarieties of normal cryptogroups. Each quasivariety studied is characterized in several ways.     

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 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.     

Divergence of Spherical General Terms of Double Fourier Series

We prove the following theorem: For arbitrary there exists a nonnegative function such that and

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.