Suppose g is a square integrable function on the real line. The principal shift invariant space, , generated by g is the closure of the span of the system
B ={g(.-k): k an integer}. These spaces are most important in many areas of Analysis. This is particulrly true in the theory of Wavelets. We begin by describing a very simple method for obtaining the basic properties of and the systems B.
The systems obtained by applying, in addition to the integral translations, also the integral modulations (these are the multiplication of a function by exp(-2pinx)) are known as the Gabor systems. By using the Zak transform we show how the same methods can be used to study the basic properties of the Gabor systems and their span.
We will define the Zak transform and explain all this
in a very simple way that will be easily understood by all who know only a "smidgeon" of mathematics. A bit more challenging will be the explanation how all this can be extended to general locally compact abelian groups and their duals.
This is joint work with E. Hernandez, H. Sikic and E. N.
Wilson.