Diametral Dimension of some pseudoconvex multiscale spaces

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 they are always pseudoconvex. We tackle here 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.