Frames

The concept of a (discrete) frame was first introduced in 1952, by Duffin and Schaeffer, who were working on non-harmonic analysis (See [4].) Their work on (discrete) frames was motivated by their observation that in some cases the sequence $\{e^{i \alpha_n t}\} \subseteq L^2(-\pi,\pi)$ has properties similar to those of an orthonormal basis. The concept of discrete frame was generalized to what is called general frame. Also, the concept of a frame in Hilbert space was extended to some Banach spaces by Grochenig. See [5] for the basic theory of frames developed by this mathematician.

Note: In this thesis, we shall consider only discrete frames in Hilbert space, but we will drop the word discrete.

Definition:

Let $H$ be a Hilbert space, and let $\{g_n\}$ be a subset of $H$. Then $\{g_n\}$ is said to be a frame of $H$ if there exist two positive finite constants, $\alpha$ and $\beta$, satisfying

$\alpha \Vert f\Vert^2 \leq \sum_n \vert\langle f,g_n \rangle\vert^2 \leq \beta \Vert f\Vert^2$, $\forall f \in H$.

The two constants $\alpha$ and $\beta$ are called frame bounds.

The frame is said to be $exact$ if any proper subset of it is not a frame. The frame is said to be tight if $\alpha=\beta$.

Definition: Let $\{$$g_n$$\}$ be a frame for the Hilbert space $H$, then the corresponding frame operator is defined by

$Sf=\sum_{n}<f,g_n>g_n,$    for all $f \in H$.

Theorem 0.1.1   [2] Let $\{g_n\}$ be a subset of the Hilbert space $H$ and let the operator $S: H \longrightarrow H$ be defined as

$Sf=\sum_n<f,g_n>g_n$, $\forall f \in H$.

Then $\{g_n\}$ is a frame for $H$ if and only if $S$ is a well-defined surjective operator.

Frames can be considered as generalized bases. In fact, frames and orthonormal bases share many properties. For example, we can use some of the sampling theorems of frames (like the first theorem, part (iv), which we have below) to reconstruct any function of any separable Hilbert space. Moreover, there are frames for $L^2(\mathbb{R})$ (as it is the case for orthonormal bases) which can be generated from a single function by translation and modulation. Such frames are called the Weyl$\verb+-+$Heisenberg coherent states (aka the Weyl$\verb+-+$Heisenberg frames, the discrete windowed Fourier transform functions, the short-time Fourier transform functions, the Gabor frame.) The Weyl$\verb+-+$Heisenberg coherent states are used in physics. And also there are frames (as it is the case for orthonormal bases) which can be generated from a single function by translation and dilation. Such frames are the most common ones and they are called wavelets. The generating function for Weyl$\verb+-+$Heisenberg coherent states or for wavelets is called the mother or analyzing wavelet. Both the Weyl$\verb+-+$Heisenberg coherent states and wavelets play an important role in signal analysis.

We shall use the following standard results about frames. The first two are taken from [13, p. 262] and the rest are taken from [6, p. 635 ff].

1. Every orthonormal basis is an exact tight frame with frame bounds $\alpha=\beta=1$.

2. If $\{g_n\}$ is a tight frame with frame bound $\alpha=1$, and if $\Vert g_n\Vert=1$ for all $n$, then $\{g_n\}$ is an orthonormal basis for $H$.

3. Frames are complete.

4. An inexact frame cannot be a basis.

5. If $\{g_n\}$ is a frame for a Hilbert space $H$ and if $f$ is any element of $H$, then $\sum_n \vert<f,g_n>\vert^2$ converges absolutely and, hence, unconditionally. This means every rearrangement of $\{g_n\}$ is also a frame.

6. In Hilbert spaces, all bounded unconditional bases are equivalent to orthonormal bases.

7. There exist frames for which certain proper subsets are frames. This represents another difference between frames and bases. In fact, it represents another flexibility of frames over basis.

8. Neither of tightness or exactness implies the other (see [6, p. 635], or [13, p. 262])

Notice that since frames are complete, then we can generate an orthonormal basis from any orthogonal frame by normalizing each element (assuming zero is not an element of the frame.)

Now let us state the following theorem which includes very important facts about frames.

Theorem 0.1.2   [13, p. 263] Let $H$ be a Hilbert space and let $\{g_n\}$ be a frame for $H$ with frame bounds $\alpha$ and $\beta$. Then

(i) The frame operator $Sf=\sum_n<f,g_n>g_n$ is a bounded linear operator on $H$ with $\alpha I \leq S \leq \beta I$.

(ii) $S$ is invertible with $\beta^{-1}I \leq S^{-1} \leq \alpha^{-1}I$. Moreover, $S^{-1}$ is a positive operator; hence it is self-adjoint.

(iii) $\{S^{-1}g_n\}$ is a frame (called the dual frame of $\{g_n\}$ with frame bounds $\beta^{-1}$, $\alpha^{-1}$.

(iv) Every $f \in H$ can be written in the form

$f=\sum_n <f,S^{-1}g_n>g_n=\sum_n <f,g_n>S^{-1}g_n$

(v) In addition, if $\{g_n\}$ is an exact frame, then $\{g_n\}$ and $\{S^{-1}g_n\}$ are biorthonormal.

Notice that the frame operator $S$ is also positive and self-adjoint.

Theorem 0.1.3   [13, p. 261] In a separable Hilbert space $H$, the following are equivalent:

(a) $\{g_n\}$ is an exact frame for $H$.

(b) $\{g_n\}$ is a bounded unconditional basis of $H$.

(c) $\{g_n\}$ is a Riesz basis of $H$.

Theorem 0.1.4   [13, p. 270] Let $g \in L^2(\mathbb{R})$ be such that $0 < \alpha \leq G(x) \leq \beta < \infty$ a.e. $\mathbb{R}$, where the function G is defined by $G(x)=\sum_n\vert g(x-na)\vert^2$. Also, assume that g is of compact support and $\operatorname{supp}(g) \subseteq I \subseteq \mathbb{R}$, where I is an interval of length $1/b$. Then $\{E_{b_m}T_{n_a}g\}$ is a frame for $L^2(\mathbb{R})$ with frame bounds $\alpha/b$, $\beta/b$. Moreover, for any $f \in L^2(\mathbb{R})$, we have

$$Sf(x)= (1/b)f(x)G(x)$$    and $$S^{-1}f(x)=bf(x)/G(x)$$

where $S$ is the frame operator.

Theorem 0.1.5   [13, p. 282] Let g be a $\sigma$-band-limited function and assume that $\{$$a_n$$\}$,$\{$$b_n$$\}$ $\subseteq \mathbb{R}$ are such that $\{$ $E_{a_n(x)}$$\}$ is a frame for $L^2[-\sigma,\sigma]$, and such that $0 < \alpha \leq G(x) \leq \beta < \infty$ a.e. $\mathbb{R}$, where the function G is defined by $G(x)=\sum_m\vert\hat{g}(x-b_m)\vert^2$. Then $\{$ $E_{a_n}T_{b_m}g$$\}$ is a frame for $L^2(\mathbb{R})$.

Corollary 0.1.6   Under the hypothesis of the last theorem, if $a_n=n a$ and $a=\frac{1}{2\sigma}$, then for any $f \in L^2(\mathbb{R})$

$S^{-1}\hat{f}=\frac{1}{2\sigma}\frac{\hat{f}}{G}$

$S^{-1}(E_{na}T_{b_m}\hat{g})=\frac{1}{2\sigma}\frac{E_{na}T_{b_m}\hat{g}}{G}$

$E_{na}T_{b_m}S^{-1}\hat{g}=\frac{1}{2\sigma}\frac{E_{na}T_{b_m}\hat{g}}{T_{b_m}G}$

Theorem 0.1.7   Let H be a Hilbert space and let $A:H \longrightarrow H$ be a linear bounded and compact operator and assume that $\{$$\{e_n\}$$\}$ $_{n=1}^{\infty}$ is an orthonormal sequence in H. Then $Ae_n \longrightarrow 0$ in H.

For background on the material of this chapter, we refer the reader to [13] and [14].