Fine continuity on metric spaces

We obtain pointwise estimates for solutions of obstacle problems on metric measure spaces and prove that p-superharmonic functions are p-finely continuous. Consequently, we show that p-quasicontinuous functions are p-finely continuous at p-quasievery point. As a byproduct, we obtain the sufficiency part of the Wiener criterion in metric spaces without the assumption of linear local connectedness. The author was supported by the Swedish Research Council.     

Some new classes of Banach-Mackey spaces

Some weakenings of property (K) of Antosik for locally convex spaces are introduced: local property (K) and, for spaces with Schauder-type decompositions, two variants of property (K) defined in terms of block- and tail-sequences. It is shown that if a space enjoys any of these new properties, then it is Banach-Mackey. An application to the barrelledness of the spaces of Pettis integrable functions is given, and examples are provided to distinguish between the various K-type properties.     

On spherical expansions of zonal functions on Euclidean spheres

This paper describes a new approach to the problem of computing spherical expansions of zonal functions on Euclidean spheres. We derive an explicit formula for the coefficients of the expansion expressing them in terms of the Taylor coefficients of the profile function rather than (as done usually) in terms of its integrals against Gegenbauer polynomials. Our proof of this result is based on a polynomial identity equivalent to the canonical decomposition of homogeneous polynomials and uses only basic properties of this decomposition together with simple facts concerning zonal harmonic polynomials. As corollaries, we obtain direct and apparently new derivations of the so-called plane wave expansion and of the expansion of the Poisson kernel for the unit ball. Received: 26 January 2007     

Some properties of generalized multiple Hermite polynomials

The purpose of this paper is to introduce and discuss a more general class of multiple Hermite polynomials. In this work, the explicit forms, operational formulas and a recurrence relation are obtained. Furthermore, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple Hermite polynomials.     

A generalized inductive limit strict topology on the space of bounded continuous functions

A generalized inductive limit strict topology β_∞ is defined on C_b(X, E), the space of all bounded, continuous functions from a zero-dimensional Hausdorff space X into a locally -convex space E, where is a field with a nontrivial and nonarchimedean valuation, for which is a complete ultrametric space. Many properties of the topology β_∞ are proved and the dual of (C_b (X, E), β_∞) is studied.     

Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions

Multiplication operators in weighted Banach (and locally convex) spaces of functions holomorphic in the unit disc are well known. In this note we investigate the connection between power boundedness, mean ergodicity and uniform mean ergodicity of such operators. Received: 13 October 2008, Revised: 18 November 2008     

Unconditional bases and the order structure

The paper deals with an analytic characterization of unconditional bases, a characterization of the unconditional character of a Schauder base for the associated cone and a number of counterexamples supporting various conditions appearing in the theorems.     

The structure of spaces of quasianalytic functions of Roumieu type

It is shown that spaces of quasianalytic ultradifferentiable functions of Roumieu type ℰ_{w}(Ω), on an open convex set , satisfy some new (Ω) -type linear topological invariants. Some consequences for the splitting of short exact sequences of these spaces as well as for the structure of the spaces are derived. In particular, Fréchet quotients of ℰ_{w}(Ω) have property (), while dual Fréchet quotients have property () of Vogt. The work of P. Domański was supported by Committee of Scientific Research (KBN), Poland, grant P03A 022 25.     

Integrals involving Gegenbauer and Hermite polynomials on the imaginary axis

Summary The integral _- [C _2n (it)]^–2(1+t ²)^–-1/2 dt is evaluated for > –1/2 whereC _2n is the Gegenbauer polynomial of degree 2n. Letting gives the value _- [H _2n (it)]^–2 e ^{1-1/2t 2} dt involving the Hermite polynomialH _2n of degree 2n. The result is obtained using Gegenbauer functions of the second kind.     

Sobolev spaces on an arbitrary metric space

We define Sobolev space W ^1,p for 1<p on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.This work is supported by KBN grant no. 2 1057 91 01     

The evaluation identification in function spaces

A study is made of the natural function which maps each point x of a space X to the evaluation function e_x:Y^x→Y defined by $\text{e}{}_{\mathrm{x}}\text{(?)=?(x)}$. A consequence of the results is that βX and υX can both be considered as subspaces of spaces of continous functions from appropriate domain spaces into I or R, respectively.     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

Widths of embeddings in function spaces

We study the approximation, Gelfand and Kolmogorov numbers of embeddings in function spaces of Besov and Triebel-Lizorkin type. Our aim here is to provide sharp estimates in several cases left open in the literature and give a complete overview of the known results. We also add some historical remarks.     

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).