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.