On the extension of the Żak transform
Ahmed I. Zayed, Piotr Mikusiński · Methods and Applications of Analysis · 1995
The Zak transform, which is a unitary transformation of L 2 (3ft) onto L 2 (Q) where Q is the unit square, has been extended to larger spaces of functions and generalized functions by many people, including A. J. E. M. Janssen.In this paper, we present a new extension of the Zak transform.The novelty of our approach is that it extends the Zak transform by using its algebraic and convolution structure instead of using its analytic properties, which are not easy to deal with.Our extension produces a topological isomorphism between the domain and the range of the transform.Moreover, it enables us to use the Zak transform in situations where other extensions do not apply.