Universal discretization

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Discretization and Anti-discretization of Rearrangement-invariant Norms

Abstract We develop a new method of discretization and anti-discretization of weighted inequalities which we apply to norms in classical Lorentz spaces and to spaces endowed with the so-called Hilbert norm. Main applications of our results include new integral conditions characterizing embeddings Γp(v) ↪→ Γq(w) and Γp(v) ↪→ Λq(w) and an integral characterization of the associate space to Γp(v),...

FLOWS AND UNIVERSAL COMPACTIFICATIONS

The main purpose of this paper is to establish a relation between universality of certain P-compactifications of a semitopological semigroup and their corresponding enveloping semigroups. In particular, we show that if we take P to be the property that the enveloping semigroup of a compactification of a semitopological semigroup s is left simple, a group, or the trivial singleton semigroup, t...

Discretization of Rational Data

Frequently one wants to extend the use of a classification method that in principle requires records with True/False values, so that records with rational numbers can be processed. In such cases, the rational numbers must first be replaced by True/False values before the method may be applied. In other cases, a classification method in principle can process records with rational numbers directl...

Bourgain’s Discretization Theorem

Bourgain’s discretization theorem asserts that there exists a universal constant C ∈ (0,∞) with the following property. Let X,Y be Banach spaces with dimX = n. Fix D ∈ (1,∞) and set δ = e−nCn . Assume that N is a δ-net in the unit ball of X and that N admits a bi-Lipschitz embedding into Y with distortion at most D. Then the entire space X admits a bi-Lipschitz embedding into Y with distortion ...