Given a collection of functions \(F = \{f_1, \ldots, f_m\} \subset L^2(\mathbb{R})\), this talk focuses on finding a shift-invariant space \(V\) with \(n\) generators \(\{\phi_1, \ldots, \phi_n\}\) that best approximates the functions in \(F\). The classical Eckart–Young theorem employs the singular value decomposition (SVD) to solve an analogous problem in finite-dimensional settings.
The key idea here is to reformulate the problem as an uncountable family of finite-dimensional problems, each of which can be addressed using an appropriate extension of the Eckart–Young theorem and how these finite-dimensional solutions can be combined and transformed to obtain the optimal shift-invariant space corresponding to the original problem. Furthermore, I will present the construction of a Parseval frame associated with this optimal space.
