Diametral dimension of some pseudoconvex multiscale spaces

Jean-Marie Aubry¹ ; Françoise Bastin²

¹ Lab. Anal. Math. Appl. (UMR CNRS 8050) Université Paris-Est 61 av. du Général de Gaulle F-94010 Créteil, France
² Département de Mathématique (B37) Université de Liège Grande Traverse, 12 B-4000 Liège, Belgique

Studia Mathematica, Tome 197 (2010) no. 1, pp. 27-42

Résumé

Stemming from the study of signals via wavelet coefficients, the spaces $S^{\nu }$ are complete metrizable and separable topological vector spaces, parametrized by a function $\nu $, whose elements are sequences indexed by a binary tree. Several papers were devoted to their basic topology; recently it was also shown that depending on $\nu $, $S^{\nu }$ may be locally convex, locally $p$-convex for some $p>0$, or not at all, but under a minor condition these spaces are always pseudoconvex. We deal with some more sophisticated properties: their diametral dimensions show that they are Schwartz but not nuclear spaces. Moreover, Ligaud's example of a Schwartz pseudoconvex non-$p$-convex space is actually a particular case of $S^{\nu }$.

Détail

DOI : 10.4064/sm197-1-3

Keywords: stemming study signals via wavelet coefficients spaces complete metrizable separable topological vector spaces parametrized function whose elements sequences indexed binary tree several papers devoted their basic topology recently shown depending may locally convex locally p convex under minor condition these spaces always pseudoconvex sophisticated properties their diametral dimensions schwartz nuclear spaces moreover ligauds example schwartz pseudoconvex non p convex space actually particular