Frames in Hilbert Spaces

Abstract

The main feature of a basis $\{f_{k}\}_{k=1}^{\infty }$ in a Hilbert space $\mathcal{H}$ is that every $f \in \mathcal{H}$ can be represented as a superposition of the elements f _k in the basis:

$$\displaystyle\begin{array}{rcl} f =\sum _{ k=1}^{\infty }c_{ k}(f)f_{k}.& &{}\end{array}$$

(5.1)

The coefficients c _k (f) are unique. We now introduce the concept of frames. A frame is also a sequence of elements $\{f_{k}\}_{k=1}^{\infty }$ in $\mathcal{H}$, which allows every $f \in \mathcal{H}$ to be written as in ( 5.1). However, the corresponding coefficients are not necessarily unique. Thus a frame might not be a basis; arguments for generalizing the basis concept were given in Chapter 4