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Józef Marcinkiewicz: analysis and probability

N. H. Bingham — 2011

Banach Center Publications

We briefly review Marcinkiewicz's work, on analysis, on probability, and on the interplay between the two. Our emphasis is on the continuing vitality of Marcinkiewicz's work, as evidenced by its influence on the standard works. What is striking is how many of the themes that Marcinkiewicz studied (alone, or with Zygmund) are very much alive today. What this demonstrates is that Marcinkiewicz and Zygmund, as well as having extraordinary mathematical ability, also had excellent mathematical taste.

Normed versus topological groups: Dichotomy and duality

The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [], and embracing its classical univariate counterpart (cf. []) as well as fragmentary multivariate (mostly Euclidean) theories (eg [], [], []), are groups with a right-invariant metric carrying flows. Following the vector paradigm, they are best seen as normed groups That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite...

Beyond Lebesgue and Baire II: Bitopology and measure-category duality

N. H. BinghamA. J. Ostaszewski — 2010

Colloquium Mathematicae

We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density topologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on infinitary combinatorics due to Kestelman and to Borwein and Ditor. We hence give a unified proof of the measure and category cases of the Uniform Convergence Theorem for slowly varying functions. We also extend results on very slowly varying functions...

Beyond Lebesgue and Baire: generic regular variation

N. H. BinghamA. J. Ostaszewski — 2009

Colloquium Mathematicae

We show that the No Trumps combinatorial property (NT), introduced for the study of the foundations of regular variation by the authors, permits a natural extension of the definition of the class of functions of regular variation, including the measurable/Baire functions to which the classical theory restricts itself. The "generic functions of regular variation" defined here characterize the maximal class of functions to which the three fundamental theorems of regular variation (Uniform Convergence,...

Very slowly varying functions. II

N. H. BinghamA. J. Ostaszewski — 2009

Colloquium Mathematicae

This paper is a sequel to papers by Ash, Erdős and Rubel, on very slowly varying functions, and by Bingham and Ostaszewski, on foundations of regular variation. We show that generalizations of the Ash-Erdős-Rubel approach-imposing growth restrictions on the function h, rather than regularity conditions such as measurability or the Baire property-lead naturally to the main result of regular variation, the Uniform Convergence Theorem.

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