Geometric Properties of Symmetric Spaces with Applications to Orlicz–Lorentz Spaces

We deal with the basic convexity properties –rotundity, and uniform, local uniform and full rotundity –- for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.     

Basic topological and geometric properties of Cesàro-Orlicz spaces

Necessary and sufficient conditions under which the Cesàro-Orlicz sequence spaceces _ϕ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro-Orlicz spacesces _ϕ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements inces _ϕ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spacesces _ϕ are given.     

Algebraic K-theory of non-linear projective spaces

In the spirit of “The Fundamental Theorem for the algebraic K-theory of spaces: I” (J. Pure Appl. Algebra 160 (2001) 21–52) we introduce a category of sheaves of topological spaces on n-dimensional projective space and present a calculation of its K-theory, a “non-linear” analogue of Quillen's isomorphism K_i(P_Rⁿ)₀ⁿK_i(R).     

α-Stable characterization of Banach spaces (1 < α < 2)

This work characterizes some subclasses of α-stable (0 < α < 1) Banach spaces in terms of the extendibility to Radon laws of certain α-stable cylinder measures. These result extend the work of S. Chobanian and V. Tarieladze (J. Multivar. Anal.7, 183–203 (1977)). For these spaces it is shown that every Radon stable measure is the continuous image of a stable measure on a suitable L_β space with β = α(1 − α)⁻¹. The latter result extends some work of Garling (Ann. Probab.4, 600–611 (1976)) and Jain (Proceedings, Symposia in Pure Math. XXXI, p. 55–65, Amer. Math. Soc., Providence, R.I.).     

Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

This paper builds upon the L_p-stability results for discrete orthogonal projections on the spaces S_h of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L₂-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.     

Embeddings in Spaces of Lipschitz Type, Entropy and Approximation Numbers, and Applications

We consider the embeddings of certain Besov and Triebel–Lizorkin spaces in spaces of Lipschitz type. The prototype of such embeddings arises from the result of H. Brézis and S. Wainger (1980, Comm. Partial Differential Equations5, 773–789) about the “almost” Lipschitz continuity of elements of the Sobolev spaces H^1+n/p_p( ⁿ) when 1<p<∞. Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings. The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators.     

Complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm

In this paper, two equivalent definitions of complex strongly extreme points in general complex Banach spaces are shown. It is proved that for any Orlicz sequence space equipped with the p-Amemiya norm (1?p<∞, p is odd), complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in Orlicz sequence spaces equipped with the p-Amemiya norm are given. Criteria for complex mid-point locally uniform rotundity and complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm are also deduced.     

On trees and dual rotund norms

We give a necessary and sufficient condition for the existence of an equivalent dual rotund norm on C₀(?)^*≡?₁(?), where ? is a tree. The condition is expressed succinctly, in terms of the embeddability of ? into a particular totally ordered set, and compares very well with the analogous situation for local uniform rotundity. This resolves an open problem from Haydon's work in Asplund spaces, trees and renorming theory.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces

Generalized Orlicz–Lorentz sequence spaces λ_φ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ^λ ₂ for φ is defined in such a way that it guarantees many positive topological and geometric properties of λ_φ. The problems of the Fatou property, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space λ_φ are considered. Moreover, some embeddings between λ_φ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λ_φ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Commutators for Fourier multipliers on Besov Spaces

If T is any bounded linear operator on Besov spaces B_p^σ_j,q_j(Rⁿ)(j=0,1, and 0<σ₁<σ<σ₀), it is proved that the commutator [T,T_μ]=TT_μ−T_μT is bounded on B_p^σ,q(Rⁿ), if T_μ is a Fourier multiplier such that μ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas.     

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution

We consider independent pairs (X₁, Σ₁), (X₂, Σ₂), …, (X_n, Σ_n), where eachΣ_iis distributed according to some unknown density functiong(Σ) and, givenΣ_i=Σ,X_ihas conditional density functionq(xΣ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationX_n+1is obtained, the objective is to estimateΣ_n+1corresponding toX_n+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.     

A Wide Class of Ultrabornological Spaces of Measurable Functions

Our main result states that a bornological locally convex space having a suitable Boolean algebra of projections is ultrabornological. This general theorem, whose proof is a variation of the sliding-hump techniques used in [Díaz et al., Arch. Math. (Basel)60 (1993), 73-78; Díaz et al., Resultate Math.23 (1993), 242-250; Drewnowski el al., Proc. Amer. Math. Sec.114 (1992), 687-694; Drewnowski et al., Atti. Sem. Mat. Fis. Univ. Modena41 (1993), 317-329], is applied to prove that some non-complete normed spaces such as the spaces of Dunford, Pettis, or McShane integrable functions, as well as other interesting spaces of weakly or strongly measurable functions, are ultrabornological. We also give applications to vector-valued sequence spaces; in particular, we prove that ℓ^p{X} (1 ≤ p < ∞) is an ultrabornological DF-space when X is.     

Majorants and Minorants for Elliptic Measures on

Let μ(· ; Σ, G₁) and μ(· ; Ω, G₂) be elliptically contoured measures on ^k centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G₁, G₂). Elliptical measures v_m(·) and v_M(·), depending on (Σ, Ω, G₁, G₂), are constructed such that V_m(C) ≤ {μ(C; Σ, G₁), μ(C; Ω, G₂)} for every symmetric convex set C ^k with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems.     

Null Sequences in Coechelon Spaces

Rotundity of finite -dimensional Orlicz spaces l^?_n equipped with the Luxemburg norm is considered. It is proved that criteria for rotundity of l^?_nfor n ≥ 3 does not depend on n and are the same as the criteria for rotundity of the inhite-dimensional subspace h^? of an Orlicz sequence spacel^?. Criteria for rotundity of l^?₂ are different. Next, criteria for exposed points, (H)- points, strongly exposed points and LUR- points of the unit sphere of l^? and of its subspace h^? are given.     

Rough Convergence and Chebyshev Centers in Banach Spaces

By means of rough convergence, we introduce two new geometric properties in Banach spaces and relate them to Chebyshev centers and some well-known classical properties, such as Kalton's M property or Garkavi's uniform rotundity in every direction.     

Strict convexity of sequence Orlicz-Musielak spaces with Orlicz norm

H. Milnes gave in (Pacific J. Math. 18 (1957), 1451–1483) a criterion for strict convexity of Orlicz spaces with respect to the so called Orlicz norm, in the case of nonatomic measure and a usual Young function. Here there are presented necessary and sufficient conditions for strict convexity of Orlicz-Musielak spaces (J. Musielak and W. Orlicz, Studia Math. 18 (1957), 49–65) with Orlicz norm in the case of purely atomic measure. For sequence Orlicz-Musielak spaces with Luxemburg norm, such a criterion is given in (A. Kami ska, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 29 (1981), 137–144).     

Laws of large numbers for classes of functions

Let (X, , P) be a probability space and _n, n ≥ 1, a sequence of classes of measurable complex-valued functions on (X, , P). Under a weak metric entropy condition on _n and sup {g_∞: g _n}, Glivenko-Cantelli theorems are established for the classes _n with respect to the probability measure P; i.e., lim_{n → ∞} sup_{g n} ∫ g(dP_n − dP) = 0 a.s. The result is applied to kernel density estimation and a law of the logarithm is derived for the maximal deviation between a kernel density estimator and its expected value, improving upon and generalizing the recent results of W. Stute (Ann. Probab. 10 (1982), 414–422). This result is also used to derive improved rates of uniform convergence for the empirical characteristic function.