Resumen de Principal shift-invariant spaces with extra invariance nearest to observed data

Akram Aldroubi, Ilya A. Krishtal, Romain Tessera, Haichao Wang

• Given a set of functions {\mathcal{F}=\{f_1, \dots, f_m\}\subset L^2(\mathbb{R}),} we construct a principal shift-invariant space V nearest to {\mathcal{F}} in the sense that V minimizes the expression \sum_{i=1}^{m}\|f_i-P_{V}f_i\|^2, among all the principal shift-invariant spaces with an orthonormal generator which are also translation invariant, or among all the principal shift-invariant spaces with an orthonormal generator which are also {\frac{1}{n}\mathbb{Z}} -invariant for some fixed {n\in\mathbb{N}}.