On Pointwise Product Vector Measure Duality

This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

Abstract relative Gabor transforms over canonical homogeneous spaces of semidirect product groups with Abelian normal factor

This paper introduces a unified approach to the abstract notion of relative Gabor transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let [Formula: see text] be a locally compact group, [Formula: see text] be a locally compact Abelian (LCA) group, and [Formula: see text] be a continuous homomorphism. Let [Formula: see text] be the semi-direct product of [Formula: see text] and [Formula: see text] with respect to [Formula: see text], [Formula: see text] be the canonical homogeneous space of [Formula: see text], and [Formula: see text] be the canonical relatively invariant measure on [Formula: see text]. Then we present a unified harmonic analysis approach to the theoretical aspects of the notion of relative Gabor transform over the Hilbert function space [Formula: see text].

On a quadratic integral inequality

SynopsisThe inequality considered iswhere p and q are given real-valued coefficients on the interval [a, b), with b ≦ ∝, of the real line; here D is a linear manifold of the Hilbert function space L2(a, b), and μ is a real number characterised in terms of the spectrum of a uniquely determined self-adjoint differential operator in L2(a, b).

16.—Square Integrable Solutions of Perturbed Linear Differential Equations

SynopsisThis paper is concerned with solutions of the ordinary differential equationwhere ℒ is a real formally self-adjoint, linear differential expression of order 2n, and the perturbed term f satisfiesfor some σ∈[0, 1]. Here λ(·) is locally integrable on [0,∞).In particular it is shown, under circumstances detailed in the text, that (*) possesses solutions in the Hilbert function space L2(0,∞).