Probabilistic spherical Marcinkiewicz–Zygmund inequalities

Recently, norm equivalences between spherical polynomials and their sample values at scattered sites have been proved. These so-called Marcinkiewicz–Zygmund inequalities involve a parameter that characterizes the density of the sampling set and they are applicable to all polynomials whose degree does not exceed an upper bound that is determined by the density parameter. We show that if one is satisfied by norm equivalences that hold with prescribed probability only, then the upper bound for the degree of the admissible polynomials can be enlarged significantly and that then, moreover, there exist fixed sampling sets which work for polynomials of all degrees.     

Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

We find necessary density conditions for Marcinkiewicz–Zygmund inequalities and interpolation for spaces of spherical harmonics in with respect to the L^p norm. Moreover, we prove that there are no complete interpolation families for p≠2.     

An algebra containing the two-sided convolution operators

We present an intrinsically defined algebra of operators containing the right and left invariant Calderón–Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<∞). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calderón–Zygmund theory.     

On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices

We strengthen the well-known Marcinkiewicz–Zygmund law of large numbers in the case of Banach lattices. Examples of applications to empirical distributions are presented.     

On a Calderón–Zygmund commutator-type estimate

In this paper we present a Calderón–Zygmund commutator-type estimate. This estimate enables us to prove an embedding result concerning weighted function spaces.     

The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

The main aim of this paper is to prove that the maximal operator σ^* of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series is bounded from the Hardy space H_2/3 to the space weak-L_2/3.     

Approximation of smooth functions on compact two-point homogeneous spaces

Estimates of Kolmogorov n-widths and linear n-widths , (1q∞) of Sobolev's classes , (r>0, 1p∞) on compact two-point homogeneous spaces (CTPHS) are established. For part of (p,q)[1,∞]×[1,∞], sharp orders of or were obtained by Bordin et al. (J. Funct. Anal. 202(2) (2003) 307). In this paper, we obtain the sharp orders of and for all the remaining (p,q). Our proof is based on positive cubature formulas and Marcinkiewicz–Zygmund-type inequalities on CTPHS.     

Embeddings of Beppo–Levi spaces in Hölder–Zygmund spaces, and a new method for radial basis function interpolation error estimates

The Beppo–Levi native spaces which arise when using polyharmonic splines to interpolate in many space dimensions are embedded in Hölder–Zygmund spaces. Convergence rates for radial basis function interpolation are inferred in some special cases.     

Existence of Value and Saddle Point in Infinite-Dimensional Differential Games

We study two-player zero-sum differential games of finite duration in a Hilbert space. Following the Berkovitz notion of strategies, we prove the existence of value and saddle-point equilibrium. We characterize the value as the unique viscosity solution of the associated Hamilton–Jacobi–Isaacs equation using dynamic programming inequalities.     

Limit Theorems for Maxima of Sums and Renewal Processes

Let {X_i}_i=1,2,... be a sequence of i.i.d. random variables, let S_n = X₁ + ... + X_n, and let S_n a.s. We discuss necessary and sufficient conditions for the Kolmogorov and Marcinkiewicz–Zygmund type strong laws of large numbers and for the law of the iterated logarithm for renewal processes defined in two different ways. Bibliography: 16 titles.     

Specht's ratio and complementary inequalities to Golden–Thompson type inequalities on the Hadamard product

As a converse of the arithmetic–geometric mean inequality, W. Specht [Math. Z. 74 (1960) 91–98] estimated the ratio of the arithmetic mean to the geometric one. In this paper, we shall show complementary inequalities to the matricial generalization of Oppenheim's inequality and the Golden–Thompson type inequalities on the Hadamard product by T. Ando [Linear Algebra Appl. 26 (1979) 203; Linear Algebra Appl. 241–243 (1996) 105], in which Specht's ratio plays an important role. As an application, we shall obtain a complementary inequality to the Hadamard determinant inequality.     

The Marcinkiewicz–Zygmund LLN in Banach Spaces: A Generalized Martingale Approach

A result due to Gut asserts that the Marcinkiewicz–Zygmund strong law of large numbers for real-valued random variables is an amart a.s. convergence property. In this paper, a necessary and sufficient condition is given, under which that SLLN is also a quasimartingale. We also study the case of Banach-space valued r.v. and show that the scalar result remains true when the space is of suitable stable type.     

A recursive procedure to generate all cuts for 0–1 mixed integer programs

We study several ways of obtaining valid inequalities for mixed integer programs. We show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to Gomory's mixed integer cuts. We also show how all valid inequalities for mixed 0–1 programs can be generated recursively from a simple subclass of the disjunctive inequalities.The research of this author was supported by NSF Contract No. ECS-8540898.     

Geometry of Log-concave Functions and Measures

We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn–Minkowski and the Blaschke–Santaló inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman’s quotient of subspace theorem, and present a functional version of the Urysohn inequality.Mathematics Subject Classiffications (2000). 52A20, 52A40, 46B07     

Order law of large numbers of the Marcinkiewicz–Zygmund type

The order law of large numbers of the Marcinkiewicz–Zygmund type is established for random variables on Banach lattices. Similar results are also obtained for the maximum scheme.     

Equidistribution of Fekete Points on the Sphere

Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well-suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of Fekete points on the sphere. The way we proceed is by showing their connection to other arrays of points, the so-called Marcinkiewicz–Zygmund arrays and interpolating arrays, that have been studied recently.     

Lifting the facets of zero–one polytopes

We discuss a procedure to obtain facets and valid inequalities for the convex hull of the set of solutions to a general zero–one programming problem. Basically, facets and valid inequalities defined on lower dimensional subpolytopes are lifted into the space of the original problem. The procedure generalizes the previously known techniques for lifting facets in two respects. First, the general zero–one programming problem is considered rather than various special cases. Second, the procedure is exhaustive in the sense that it accounts for all the facets (valid inequalities) which are liftings of a given lower dimensional facet (valid inequality).     

Existence results for systems of vector variational-like inequalities

The purpose of this paper is to introduce and study systems of vector variational-like inequalities in Banach spaces. Under certain conditions, some existence results for systems of vector variational-like inequalities in Banach spaces are obtained by Kakutani–Fan–Glicksberg fixed point theorem.     

Wiener–Hopf Equations Technique for Quasimonotone Variational Inequalities

In this paper, we use the Wiener–Hopf equations technique to suggest and analyze new iterative methods for solving general quasimonotone variational inequalities. These new methods differ from previous known methods for solving variational inequalities.     

Sensitivity Analysis for Quasi-Variational Inequalities

In this paper, we develop a sensitivity analysis framework for quasi-variational inequalities using the Wiener–Hopf equations technique without assuming the differentiability of the given data.