Data approximation in twisted shift-invariant spaces

Abstract

Twisted convolution is a non-standard convolution which arises while transferring the convolution of the Heisenberg group to the complex plane. Under this operation of twisted convolution, \(L^{1}(\mathbb {R}^{2n})\) turns out to be a non-commutative Banach algebra. Hence the study of (twisted) shift-invariant spaces on \(\mathbb {R}^{2n}\) completely differs from the perspective of the usual shift-invariant spaces on \(\mathbb {R}^{d}\). In this paper, by considering a set of functional data \(\mathcal {F}=\{f_{1},\ldots ,f_{m}\}\) in \(L^{2}(\mathbb {R}^{2n})\), we construct a finitely generated twisted shift-invariant space \(V^{t}\) on \(\mathbb {R}^{2n}\) in such a way that the corresponding system of twisted translates of generators form a Parseval frame sequence and show that it gives the best approximation for a given data, in the sense of least square error. We also find the error of approximation of \(\mathcal {F}\) by \(V^{t}\). Finally, we illustrate this theory with an example.