arXiv subjects

A. J. Ostaszewski

Publications and source records attributed to A. J. Ostaszewski.

23 records · Page 2Linked to original sources

arXiv2016

Effros, Baire, Steinhaus and Non-Separability

We give a short proof of an improved version of the Effros Open Mapping Principle via a shift-compactness theorem (also with a short proof), involving `sequential analysis' rather than separability, deducing it from the Baire property in a general Baire-space setting (rather than under topological completeness). It is applicable to absolutely-analytic normed groups (which include complete metrizable topological groups), and via a Steinhaus-type Sum-set Theorem (also a consequence of the shift-compactness theorem) includes the classical Open Mapping Theorem (separable or otherwise).

math.GN↗

arXiv2014

Cauchy's functional equation and extensions: Goldie's equation and inequality, the Gołąb-Schinzel equation and Beurling's equation

The Cauchy functional equation is not only the most important single functional equation, it is also central to regular variation. Classical Karamata regular variation involves a functional equation and inequality due to Goldie; we study this, and its counterpart in Beurling regular variation, together with the related Gołąb-Schinzel equation.

math.CA↗

arXiv2013

Uniformity and self-neglecting functions: II. Beurling Regular Variation and the class Γ

Beurling slow variation is generalized to Beurling regular variation. A Uniform Convergence Theorem, not previously known, is proved for those functions of this class that are measurable or have the Baire property. This permits their characterization and representation. This extends the gamma class of de Haan theory studied earlier.

math.CA↗

arXiv2013

Uniformity and self-neglecting functions

We relax the continuity assumption in Bloom's uniform convergence theorem for Beurling slowly varying functions ϕ. We assume that ϕhas the Darboux property, and obtain results for ϕmeasurable or having the Baire property.

math.CA↗

arXiv2010

Kingman, category and combinatorics

Kingman's Theorem on skeleton limits---passing from limits as $n\to \infty $ along $nh$ ($n\in \mathbb{N}$) for enough $h>0$ to limits as $t\to \infty $ for $t\in \mathbb{R}$---is generalized to a Baire/measurable setting via a topological approach. We explore its affinity with a combinatorial theorem due to Kestelman and to Borwein and Ditor, and another due to Bergelson, Hindman and Weiss. As applications, a theory of `rational' skeletons akin to Kingman's integer skeletons, and more appropriate to a measurable setting, is developed, and two combinatorial results in the spirit of van der Waerden's celebrated theorem on arithmetic progressions are given.

math.CA↗