Principal shift-invariant spaces with extra invariance nearest to observed data
The principal shift-invariant (PSI) space generated by \(\phi \in L^2(\mathbb R)\) is the space \[ V(\phi) = \{ \sum_{k \in \mathbb Z} c_k \, \phi(\cdot -k): (c_k) \in \ell^2\}. \] Shift-invariant spaces are widely used in approximation theory, numerical analysis, wavelet theory, etc. Of particular interest are those PSI spaces with an orthonormal generator which also satisfy some additional invariance property, such as translation invariance or invariance with respect to \(\frac 1n {\mathbb Z}\) translations, for some \(n \in \mathbb N\). In a prior work, the authors have shown that, given a set of functions \(F =\{f_1, \dots. f_n\} \subset L^2(\mathbb R)\), there exists a PSI space \(V(\phi) \) with an orthonormal generator which is the nearest to \(F\) in the least squares sense, that is, \[ V(\phi) = \text{argmin}_{\, V' \in \mathcal V} \sum_{i=1}^m || f_i - P_{V'} f_i||^2, \] where \({\mathcal V}\) is the set of all PSI spaces of \(L^2(\mathbb R)\) with an orthonormal generator and \(P_{V'}\) is the orthogonal projection on \(V'\). The main result of this paper is the following refinement of the above observation. Namely, given a set of functions \(F =\{f_1, \dots. f_n\} \subset L^2(\mathbb R)\), the authors show how to construct a PSI space \(V(\phi)\) with an orthonormal generator which is the nearest to \(F\) in the least squares sense, where \(V(\phi)\) also satisfies translation invariance or invariance with respect to \(\frac 1n {\mathbb Z}\) translations, for some fixed \(n \in \mathbb N\).
scientific article; zbMATH DE number 6175357
scientific article