Some properties of rational approximations of degree (k, 1) in the hardy spaceH 2(D)

We prove that the well-known interpolation conditions for rational approximations with free poles are not sufficient for finding a rational function of the least deviation. For rational approximations of degree (k, 1), we establish that these interpolation conditions are equivalent to the assertion that the interpolation pointc is a stationary point of the functionk(c) defined as the squared deviation off from the subspace of rational functions with numerator of degree k and with a given pole 1/¯c. For any positive integersk ands, we construct a functiong H₂(D) such thatR _k ,1(g)=R _k +s,1(g) > 0. whereR _k ,1(g) is the least deviation ofg from the class of rational function of degree (k, 1).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 251–259, August, 1998.The author is keenly grateful to N. S. Vyacheslavov, E. P. Dolzhenko, and V. G. Zinov for useful discussions.     

Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side was carried over by
Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of is a necessary condition here if is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in by taking coordinate functionals).

     

Norms of embeddings of logarithmic Bessel potential spaces

Let be a subset of with finite volume, let and let be a Young function with for large . We show that the norm on the Orlicz space is equivalent to

We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in which are sharp in their dependences on provided that is large enough.

     

On Convex Limit Sets and Brownian Motion

We prove limsup results for nonnegative functionals of convex sets determined by normalized Brownian paths in Banach spaces. This continues the interesting investigation of D. Khoshnevisan into this area, and relates to some classical unsolved isoperimetric problems for the convex hull of curves in ^d. Section 4 contains the solution of a problem similar to these classical problems.     

Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The -Theory

We prove that the exact reconstruction of a function from its samples on any ``sufficiently dense" sampling set can be obtained, as long as is known to belong to a large class of spline-like spaces in . Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.

     

RUC-Bases in Orlicz and Lorentz Operator Spaces

If is an RUC-basis in somecouple of non-commutative L _p-spaces, then is an RUC-basic sequence in any non-commutative Orlicz or Lorentz space which is an interpolation space for this couple.     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

Descent Theory and Morita Theory for Ultrametric Banach Modules

In this paper we consider ultrametric Banach modules over commutative ultrametric Banach algebras with unit. We study the descent problem along a morphism f: R S of such algebras and show that descent morphisms coincide with weak retracts. We give further conditions for having an effective descent morphism or for having a Morita equivalence between the corresponding categories of ultrametric Banach modules.     

Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmooth analysis. Then we study several versions of smooth variational principles in Asplund spaces, provide necessary and sufficient conditions for the validity of such principles, and establish their relationships with certain subdifferential properties of lower semicontinuous functions.