What are examples of frame operators?

Question

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in H$: $$A\|f\|^2\leq\sum_{n=1}^\infty |(f,f_n)|^2\leq B \|f\|^2, \ \ \ (\ast)$$ An operator $T$ on $H$ is said "frame operator" of frame $\{f_1,f_2,...\}$, if $$Tf=\sum_{n=1}^\infty (f,f_n) f_n$$ By this the $(\ast)$ becomes, $$A\|f\|^2\leq (Tf,f)\leq B \|f\|^2$$

This is all true in theory, but I've never seen a practical example of frame and, above all frame operator. I emphasize: especially examples of frame operators. Because while I can imagine an example of frame (a set of generators linearly dependent), I can not find an example of frame operator.

They are like answers containing examples and references (books, papers, websites).

Thanks in advance.

Answer 1

An example of a frame operator depends of course on the frame. For a frame $(f_n)_{n \in \mathbb{N}}$ in a Hilbert spaces $\mathcal{H}$, the frame operator $S : \mathcal{H} \to \mathcal{H}$ is, as you mentioned, just defined as $$ S f = \sum_{n \in \mathbb{N}} \langle f, f_n \rangle f_n. \quad \quad (*)$$ To my knowledge, this is the only form in which you can write the frame operator for abstract frames. To obtain "an example of a frame operator", you just plug in the elements $f_n$ of the frame.

The situation is different for "concrete" frames. For example, a Gabor frame for $L^2 (\mathbb{R})$ is a collection of the form $\{e^{2 \pi i b x} g(x-ak)\}_{k,n \in \mathbb{Z}}$, where $a, b \in \mathbb{R}^+$ and $g \in L^2 (\mathbb{R})$, that forms a frame for $L^2 (\mathbb{R})$. For such a frame, the frame operator becomes, by definition, $$ [S f](x) = \sum_{n \in \mathbb{Z}} \sum_{n \in \mathbb{Z}} \bigg(\int_{\mathbb{R}} f(x) e^{-2 \pi i b x} \overline{g(x-ak)} d\mu(x) \bigg) e^{2 \pi i b x} g(x-ak). $$ One can show, by assuming several conditions, that this frame operator can be re-written as $$[Sf](x) = \frac{1}{b} \sum_{n \in \mathbb{Z}} f(x - \frac{n}{b}) \sum_{k \in \mathbb{Z}} g(x-ak) \overline{g(x - ak - \frac{n}{b})}, $$ which is known as the Walnut representation in Gabor analysis.

An excellent book on frame theory is An Introduction to Frames and Riesz Bases by Ole Christensen. The first half of that book considers "abstract" frames in general Hilbert spaces; the second half of that book considers "concrete" frames in mainly the Hilbert space $L^2 (\mathbb{R})$.

Answer 2

Here's an easy finite-dimensional example:

Consider the frame $\{u_1\equiv (1,0)^T, u_2\equiv (0,1)^T, u_3\equiv(1,1)^T\}\subset\mathbb R^2$. The associated frame operator $S$ can be written in dyadic notation as $S\equiv\sum_{k=1}^3 u_k u_k^\dagger$, with $u_k^\dagger$ dual vector to $u_k$, and thus, as a matrix, $$S= \begin{pmatrix}2& 1 \\ 1 & 2\end{pmatrix}.$$ You can then for example notice how this has the eigenvalues $1,3$, which are in fact the frame bounds of the considered frame. The associated eigenvectors are $v_1\equiv \frac{1}{\sqrt2}(1,-1)^T$ and $v_3\equiv \frac{1}{\sqrt2}(1,1)^T$, respectively, and one can observe how these two vectors correspond to vectors of coefficients with norm equal to the respective eigenvalues: writing $v=\sum_{k=1}^3 c_k(v) u_k$, where $c_k(v)=\langle S^{-1}u_k,v\rangle$, we get $$\boldsymbol{c}(v_1) =\frac1{\sqrt2}\begin{pmatrix}1 \\ -1\\ 0\end{pmatrix} \implies \|\boldsymbol{c}(v_1)\|^2=1, \\ \boldsymbol{c}(v_3) =\frac1{3\sqrt2}\begin{pmatrix}1\\1\\2\end{pmatrix} \implies \|\boldsymbol{c}(v_3)\|^2=\frac13.$$ This is in line with the general fact that decomposing an eigenvector of $S$ with eigenvalue $\lambda$ as a linear combination of the frame elements gives back a vector of coefficients with norm $1/\lambda$. To see it, suppose $Sv=\lambda v$ and $(u_k)_k$ are the frame elements. Then $$c_k(v) = \langle S^{-1}u_k,v\rangle=\frac1{\lambda}\langle u_k,v\rangle\implies \|\boldsymbol c(v)\|^2 = \frac{1}{\lambda^2}\langle v,Sv\rangle = \frac{1}{\lambda}.$$