Extremal structure and Samuel compactification

Let M be a uniform space and X the Banach space of bounded and uniformly continuous functions from M into R, provided with its supremum norm.The aim of this paper is to discuss the connection between the geometry of X and the nature of M. In particular, we will prove that certain reconstructions of the unit ball of X by means of its extreme points admit a translation in terms of extension of uniformly continuous functions. We also analyze the impact of these properties on the Samuel compactification of M.     

On interpolation of interpolating sequences

Let A be a uniform algebra on the compact space X and σ a probability measure on X. We define the Hardy spaces H^P(σ) and the H^P(σ) interpolating sequences S in the p-spectrum M_p of σ. Under some structural hypotheses on (A, σ), we prove that if a sequence S ⊂ M_p is H^P(σ) interpolating, then it is H^s(σ) interpolating for s < p. In the special case of the unit ball B of ?ⁿ this answers a natural question asked in [8].     

On nonmeasurable unions

For a Polish space X and a σ-ideal I of subsets of X which has a Borel base we consider families A of sets in I with the union ?A not in I. We determine several conditions on A which imply the existence of a subfamily A^′ of A whose union ?A^′ is not in the σ-field generated by the Borel sets on X and I. Main examples are X=R and I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.     

Inner functions as improving multipliers

If X⊂Y are two classes of analytic functions in the unit disk D and θ is an inner function, θ is said to be (X,Y)-improving, if every function f∈X satisfying fθ∈Y must actually satisfy fθ∈X. This notion has been recently introduced by K.M. Dyakonov. In this paper we study the (X,Y)-improving inner functions for several pairs of spaces (X,Y). In particular, we prove that for any p∈(0,1) the (Qp,BMOA)-improving inner functions and the (Qp,B)-improving inner functions are precisely the inner functions which belong to the space Qp. Here, B is the Bloch space. We also improve some results of Dyakonov on the subject regarding Lipschitz spaces and Besov spaces.     

Numerical peak holomorphic functions on Banach spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab(BX:X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense Gδ-subset of Ab(BX:X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab(BX:X). In particular, when X=Lp(μ)(1<p<∞) or X=?₁, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense Gδ-subset of Ab(BX:X). As an application, the existence and properties of numerical boundary of Ab(BX:X) are studied. Finally, the numerical peak function in Ab(BX:X) is characterized when X=C(K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.     

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)^**.     

On holomorphic functions attaining their norms

We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon-Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k, it cannot be approximated by norm attaining polynomials with degree less than k. For , a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm.     

Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions

Let X be a p-dimensional random vector with density f(6X?θ6) where θ is an unknown location vector. For p ≥ 3, conditions on f are given for which there exist minimax estimators θ?(X) satisfying 6Xt6 · 6θ?(X) ? X6 ≤ C, where C is a known constant depending on f. (The positive part estimator is among them.) The loss function is a nondecreasing concave function of 6θ?? θ6². If θ is assumed likely to lie in a ball in $\text{R}$^p, then minimax estimators are given which shrink from the observation X outside the ball in the direction of P(X) the closest point on the surface of the ball. The amount of shrinkage depends on the distance of X from the ball.     

Representations of the Lie algebra of vector fields on a sphere

For an affine algebraic variety X we study a category of modules that admit compatible actions of both the algebra A of functions on X and the Lie algebra of vector fields on X. In particular, for the case when X is the sphere ${\mathbb{S}}^2$, we construct a set of simple modules that are finitely generated over A. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational ${\mathrm{GL}}_2$-modules.     

Reduction of variance for Gaussian densities via restriction to convex sets

Let X be a random vector with values in $\text{R}$ⁿ and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on $\text{R}$ⁿ. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a ∈ $\text{R}$ⁿ. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.     

On multiplicative perturbation of integral resolvent families

In this paper we study multiplicative perturbations for the generator of a strongly continuous integral resolvent family of bounded linear operators defined on a Banach space X. Assuming that a(t) is a creep function which satisfies a(⁰⁺)>0, we prove that if (A,a) generates an integral resolvent, then (A(I+B),a) also generates an integral resolvent for all B∈B(X,Z), where Z belongs to a class of admissible Banach spaces. In special instances of a(t) the space Z is proved to be characterized by an extended class of Favard spaces.     

On relations between γ-operations

Let expX be the power set of a non-empty set?X. A function γ:?expX→expX is said to be monotonic iff A?B?X implies γA?γB. Császár?[2] investigated relations between the monotonic functions. The purpose of the paper is to investigate some results concerning particular monotonic functions.     

Relative ranks of Lipschitz mappings on countable discrete metric spaces

Let X be a countable discrete metric space and let XX denote the family of all functions on X. In this article, we consider the problem of finding the least cardinality of a subset A of XX such that every element of XX is a finite composition of elements of A and Lipschitz functions on X. It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountable.We show that if X contains a Cauchy sequence or a sufficiently separated, in some sense, subspace, then |A|≤1. On the other hand, we give several results relating |A| to the cardinal d; defined as the minimum cardinality of a dominating family for NN. In particular, we give a condition on the metric of X under which |A|≥d holds and a further condition that implies |A|≤d. Examples satisfying both of these conditions include all subsets of Nk and the sequence of partial sums of the harmonic series with the usual euclidean metric.To conclude, we show that if X is any countable discrete subset of the real numbers R with the usual euclidean metric, then |A|=1 or almost always, in the sense of Baire category, |A|=d.     

On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology τ_F on 2 ^X has as a subbase all sets of the form {A ∈ 2 ^X :A ∩V ≠ 0}, whereV is an open subset ofX, plus all sets of the form {A ∈ 2 ^X :A ?W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for τ_F in terms of topological properties for τ. Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.     

Pseudoconvex domains spread over complex homogeneous manifolds

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X = G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.     

Rank-1 perturbations of cosine functions and semigroups

Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each B∈L(D((ω−A)^1/2),X). If A is unbounded and , then we show that there exists a rank-1 operator B∈L(D(γ(ω−A)),X) such that A+B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A+B generates a distribution semigroup for each operator B∈L(D(A),X) of rank-1, then A generates a holomorphic C₀-semigroup. If A+B generates a C₀-semigroup for each operator B∈L(D(γ(ω−A)),X) of rank-1 where 0<γ<1, then the semigroup T generated by A is differentiable and ‖T^′(t)‖=O(t−α) as t↓0 for any α>1/γ. This is an approximate converse of a perturbation theorem for this class of semigroups.     

Sharp estimates of the error of interpolation by bilinear splines for some classes of functions

On a two-point homogeneous space X, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in X. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.     

Points of Weak-Norm Continuity in the Unit Ball of Banach Spaces

Using the M-structure theory, we show that several classical function spaces and spaces of operators on them fail to have points of weak-norm continuity for the identity map on the unit ball. This gives a unified approach to several results in the literature that establish the failure of strong geometric structure in the unit ball of classical function spaces. Spaces covered by our result include the Bloch spaces, dual of the Bergman space L¹_a and spaces of operators on them, as well as the space C(T)/A, where A is the disc algebra on the unit circle T. For any unit vector f in an infinite-dimensional function algebra A we explicitly construct a sequence {f_n} in the unit ball of A that converges weakly to f but not in the norm.     

Representable Functions on the Unit Ball of a Banach Space

In this paper we treat the problem of integral representation of analytic functions over the unit ball of a complex Banach space X using the theory of abstract Wiener spaces. We define the class of representable functions on the unit ball of X and prove that this set of functions is related with the classes of integral k–homogeneous polynomials, integral holomorphic functions and also with the set of L ^p –representable functions on a Banach space.     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.