Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups

We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gr\"ochenig, Kutyniok, and Seip in the article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255 (2008) 1831-1850). To achieve this result, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of Rd.Comment: 21 pages, 1 figur

Who is mentally ill? Psychiatry and the Individual in the Interwar Period in Germany

Sixteen nations were involved in the First World War with over 65 million soldiers in active service and nearly one million soldiers suffered, according to medical documentation, from psychic consequences of war. The authors are historians who analyzed historical sources and present psychic consequences of the First World War from two aspects: on the one hand, from the viewpoint of German psychiatry, which denied the vulnerability of the human psyche; on the other hand, from the viewpoint of an individual (Hungarian politician, Loránt Hegedüs), who was an inpatient of an elegant Berlin sanatorium between 1921 and 1924 and argued critically against war. The question posed by the title, Who is mentally ill? will be examined. Norms can also be insane if they do not serve man but instead they are inhuman. How do people react to these expectations? The normative order of the state and of psychiatry expected soldiers to die a heroic death for the nation and stigmatized “weaklings” as “abnormal”. Hegedüs, the Hungarian minister of finance was faced with the issue of normality as he failed with his financial program in 1921, in other words he “collapsed” under the weight of responsibility for the nation. Researching these social phenomenon adds to our understanding in the history of Central European societies

Lorentz-Shimogaki and Boyd Theorems for weighted Lorentz spaces

We prove the Lorentz-Shimogaki and Boyd theorems for the spaces Λpu(w). As a consequence, we give the complete characterization of the strong boundedness of H on these spaces in terms of some geometric conditions on the weights u and w, whenever p > 1. For these values of p, we also give the complete solution of the weak-type boundedness of the Hardy Littlewood operator on Λpu(w).Fil: Agora, Elona. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad de Barcelona; EspañaFil: Antezana, Jorge Abel. Universidad Nacional de la Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaFil: Carro, María Jesús. Universidad de Barcelona; EspañaFil: Soria, Javier. Universidad de Barcelona; Españ

Characterization of the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces

We characterize the weak-type boundedness of the Hilbert transform $H$ on weighted Lorentz spaces $\Lambda^p_u(w)$, with $p>0$, in terms of some geometric conditions on the weights $u$ and $w$ and the weak-type boundedness of the Hardy-Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of $H$ on weighted Lebesgue spaces $L^p(u)$ and Muckenhoupt weights $A_p$, and the theory on classical Lorentz spaces $\Lambda^p(w)$ and Ari\~no Muckenhoupt weights $B_p$.Comment: 1

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