In this paper we present topology aspects in non- localization results. A well-known such no-go result in the Balian-Low theorem that states the generator of a Weyl- Heisenberg Riesz basis cannot be well time-frequency localized. More general, the statement applies to multi WH Riesz bases, or super frames as well. These results turn out to be connected to non-triviality of a complex vector bundle. Another class of problem is related to optimality of coherent approximations of stochastic signals. More specific, for a given deficit ((alpha) (beta) >1), find the best Riesz sequence generator optimal to respect to the mean square approximation error. A topological obstruction turns out to be responsible for ill-localization of the optimal generator.
Radu V. Balan, Radu V. Balan,
"Topological obstructions to localization results", Proc. SPIE 4478, Wavelets: Applications in Signal and Image Processing IX, (5 December 2001); doi: 10.1117/12.449701; https://doi.org/10.1117/12.449701