An Approximation Problem in Multiplicatively Invariant Spaces

Let $\mathcal{H}$ be Hilbert space and $(\Omega,\mu)$ a $\sigma$-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega, \mathcal{H})$ that are invariant under point-wise multiplication by functions in a fix subset of $L^{\infty}(\Omega).$ Given a finite set of data $\mathcal{F}\subseteq L^2(\Omega, \mathcal{H}),$ in this paper we prove the existence and construct an MI space $M$ that best fits $\mathcal{F}$, in the least squares sense. MI spaces are related to shift invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation invariant spaces are in correspondence with totally decomposable MI spaces.Comment: 18 pages, To appear in Contemporary Mathematic

Linear combinations of frame generators in systems of translates

A finitely generated shift invariant space $V$ is a closed subspace of $L^2(\R^d)$ that is generated by the integer translates of a finite number of functions. A set of frame generators for $V$ is a set of functions whose integer translates form a frame for $V$. In this note we give necessary and sufficient conditions in order that a minimal set of frame generators can be obtained by taking linear combinations of the given frame generators. Surprisingly the results are very different to the recently studied case when the property to be a frame is not required.Comment: 13 pages, To appear in J. Math. Anal. App

Weak-Scale Hidden Sector and Energy Transport in Fireball Models of Gamma-Ray Bursts

The annihilation of pairs of very weakly interacting particles in the neibourghood of gamma-ray sources is introduced here as a plausible mechanism to overcome the baryon load problem. This way we can explain how these very high energy gamma-ray bursts can be powered at the onset of very energetic events like supernovae (collapsars) explosions or coalescences of binary neutron stars. Our approach uses the weak-scale hidden sector models in which the Higgs sector of the standard model is extended to include a gauge singlet that only interacts with the Higgs particle. These particles would be produced either during the implosion of the red supergiant star core or at the aftermath of a neutron star binary merger. The whole energetics and timescales of the relativistic blast wave, the fireball, are reproduced.Comment: 4 pp, 1 ps fig, text revised and improve

Application of new measurement techniques and strategies to measure ammonia emissions from agricultural activities

Agriculture is the main contributor to the ammonia emissions in the Netherlands. In order to comply with the ammonia emission reduction assigned to the Netherlands, new techniques have been implemented to reduce the ammonia emissions from animal houses, and after application of slurry into the field