On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in \mathbbC \mathbb{C} are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ \mathbbR³ \mathbb{R}^3 is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ \mathbbC \mathbb{C} (when f is assumed to be in C ¹ with [`(?)] f\bar \partial {\kern 1pt}f small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

Polynomial approximation on the boundary and strictly inside

We investigate the possibility of approximating a function on a compact setK of the complex plane in such a way that the rate of approximation is almost optimal onK, and the rate inside the interior ofK is faster than on the whole ofK. We show that ifK has an external angle smaller than π at some point z_o∈δK, then geometric convergence insideK is possible only for functions that are analytic at z_o. We also consider the possibility of approximation rates of the form exp(?cn ^β), for approximation insideK, where β is related to the largest external angle ofK. It is also shown that no matter how slowly the sequence {γ _n } tends to zero, there is aK and a Lip β, β<1, functionf such that approximation insideK cannot have order {γ _n }.     

Degree of approximation of analytic functions by “near-best” polynomial approximants

We study the rate of approximation of a functionfA(K) in the interior of the compact setK by polynomials, that are close to best polynomial approximants on the whole setK. Lower and upper estimates of possible improvement of convergence (depending on the geometry ofK) are obtained.Communicated by Vilmos Totik.     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

On a Class of Hungarian Semigroups and the Factorization Theorem of Khinchin

Let G be a connected reductive Lie group and K be a maximal compact subgroup of G. We prove that the semigroup of all K-biinvariant probability measures on G is a strongly stable Hungarian semigroup. Combining with the result [see Rusza and Szekely⁽⁹⁾], we get that the factorization theorem of Khinchin holds for the aforementioned semigroup. We also prove that certain subsemigroups of K-biinvariant measures on G are Hungarian semigroups when G is a connected Lie group such that Ad G is almost algebraic and K is a maximal compact subgroup of G. We also prove a p-adic analogue of these results.     

A counterexample in the theory of best approximation

We give an example of a domain Ω with smooth boundary and with compact subsets K₁ and K₂, such that K₁ and K₂ have disjoint hulls, but such that there is no function u, harmonic on Ω, which is negative on K₁ and positive on K₂.     

On universal subsets of Banach spaces

We prove that to most of the known hypercyclic operators A on separable Banach spaces there exist compact (compact convex, compact connected) subsets K of E such that each compact (compact convex, compact connected) subset of E can be approximated with respect to Hausdorff's distance by for suitable . Received July 8, 1997, in final form October 17, 1997     

The facialQ-topology for compact convex sets

The structure of the set of closed two-sided ideals in aC*-algebraU with identity is described by means of a topology on the set _e K of extreme points of the state spaceK ofU. Recent results of Alfsen, Andersen, Combes, Perdrizet, Wils, and others have shown that such a topology can be defined on the set _e K of extreme points of an arbitrary compact convex subset of a locally convex Hausdorff topological vector space.The structure of the set of closed left ideals in aC*-algebraU with identity can also be described by means of a set of subsets of the set _e K of extreme points of its state spaceK. Akemann, Giles, and Kummer showed that this formed a more general structure than a topology which was called aq-topology. In this paper it is shown that for a reasonably wide class of compact convex subsetsK of locally convex Hausdorff topological vector spaces such aq-topology can also be defined on _e K and that it shares many of the properties of theq-topology defined forC*-algebras. The methods used depend strongly upon recent results of Alfsen and Shultz on the spectral theory of affine functions on compact convex sets.     

Pseudoconvexity in Lorentzian doubly warped products

A Lorentzian manifold M is said to be null (resp. causally) pseudoconvex if, given any compact set K in M, there exists a compact set K' in M such that any null (resp. causal) geodesic segment with both endpoints in K lies in K'. Various implications of causal and null pseudoconvexity on the geodesic structure of a Lorentzian manifold have been studied in several recent papers by Beem and Parker, Beem and Ehrlich, and Low. We provide sufficient conditions for a Lorentzian doubly warped product manifold to be null pseudoconvex. These conditions are not necessary and provide new examples of non-globally hyperbolic spacetimes which are null pseudoconvex.     

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S ²ⁿ⁺¹ is the Reeb vector field of a natural Sasakian structure on S ²ⁿ⁺¹. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction. Supported by funds of the University of Lecce and M.I.U.R.(PRIN).     

A Note on the Moment Map on Compact Kähler Manifolds

We consider compact Kähler manifolds acted on effectively by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||||² is constant if and only if K is semisimple and the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler manifold which is acted on trivially by K.     

Bounded symmetric domains and polynomial convexity

For every bounded symmetric domain D=G/K in a complex vector space E of finite dimension we determine all polynomially convex compact subsets of E that are invariant under the compact linear group K, where D is realized in the standard way as circular convex domain in E.Mathematics Subject Classification (2000): 17C50, 32E20, 32M15, 32V25     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

The Katětov property for finite degree seminormal functors

A seminormal functor kF enjoys the Katěetov property (K-property) if for every compact set X the hereditary normality of kF(X) implies the metrizability of X. We prove that every seminormal functor of finite degree n>3 enjoys the K-property. On assuming the continuum hypothesis (CH) we characterize the weight preserving seminormal functors with the K-property. We also prove that the nonmetrizable compact set constructed in [1] on assuming CH is a universal counterexample for the K-property in the class of weight preserving seminormal functors.     

Simultaneous approximation and interpolation in weighted spaces

In this paper we present a result about simultaneous approximation and interpolation in weighted spaces. It generalizes a result of Prolla in the space of continuous functionsC(X;E) whereX is a compact Hausdorff space andE is a normed space. As a consequence, we prove that simultaneous approximation and interpolation is possible from certain vector subspaces.     

On best rational approximation of analytic functions

This paper contains some theorems related to the best approximation ρ_n(f;E) to a function f in the uniform metric on a compact set by rational functions of degree at most n. We obtain results characterizing the relationship between ρ_n(f;K) and ρ_n(f;E) in the case when complements of compact sets K and E are connected, K is a subset of the interior Ω of E, and f is analytic in Ω and continuous on E.     

Pointwise convergence and radial limits of harmonic functions

This paper characterizes those real-valued functions on a compact setK in ℝ ⁿ that can be expressed as the pointwise limit of a sequence (h _m ), where each functionh _m is harmonic on some neighbourhood ofK. It also characterizes those functions on the unit sphere that can arise as the radial limit function at infinity of an entire harmonic function. Both results rely on important recent work of Lukeš et al. concerning approximation of affine Baire-one functions.     

Actions of compact groups on coherent sheaves

We consider actions of compact real Lie GroupsK on complex spacesX such that the associated reducedK-space admits a semistable quotient, e.g.X is a Stein space. We show that there is a complex spaceX ^c endowed with a holomorphic action of the universal complexificationG ofK that containsX as an openK-stable subset. As our main result, we prove that every coherentK-sheaf onX extends uniquely to a holomorphicG-sheaf onX ^c .Supported by a Heisenberg Stipendium of the Deutsche Forschungsgemeinschaft.