Overview

The general notion of a "frame" will enable us to present the continuous wavelet transform and its discretized version (to be studied later on) from a single functional-analytic viewpoint. The next two sections, 4.1 and 4.2, are essentially borrowed from [K], where this unified aspect of the two theories is described in a particularly lucid way.

To summarize the general idea in a few lines: A frame is a collection a. ? a _? ? ? I) of vectors in a Hilbert space X that is rich enough to make sure that no vector x ? X other than 0 is orthogonal to all a _?. In the infinite-dimensional case this is not so easy to guarantee. The a _? need not be linearly independent, let alone orthonormal. As a consequence, frames are in general a "redundant" collection of vectors.

4.1 Geometrical Considerations

In order to get acquainted with the proposed "framework" we consider the following situation:

Let X be a finite- dimensional complex Hilbert space: dim X ? n < ?, and assume that r vectors a ₁, ..., a _r ? X are given. The number r of these vectors should be thought of by the reader as being larger than the dimension n of the space X. With the aid of these a _j we construct the mapping

Denoting the canonical basis of ? ^r ? Y