Bases and Frames
A basis is a linearly independent set of vectors that spans a given Hilbert space such that any function in that space can be expanded as a unique combination of these linearly independent vectors.On the other hand, frames are a generalization of bases such that they may be linearly dependent and, therefore, redundant. Formal definitions follow.
Definition of Bases: Consider a set of vectors \(\{e_{k}\}_{k=1}^{\infty}\) in \(H\)
- 1. This set is a basis in \(H\) if for each value of \(f \in H \) there is a set of unique scalars \(c_{k}^{f}\) such that \[ f=\sum_{k=1}^{\infty} c_{k}^{f}e_{k} \]
- 2. The basis \(\{e_{k}\}_{k=1}^{\infty}\) is an orthonormal basis if it is an orthonormal system \[ \lt e_{k},e_{j} \gt = \delta_{k,j} \]
Orthonormal Basis properties
- 1. \(f=\sum_{k=1}^{\infty} \lt f,e_{k} \gt e_{k}\) , \(\forall f \in H \)
- 2. \(\lt f,g \gt=\sum_{k=1}^{\infty} \lt f,e_{k} \gt \lt e_{k},g \gt\) , \(\forall f,g \in H \)
- 3. \(\overline{span}\{e_{k}\}_{k=1}^{\infty}= H\)
- 4. \(\sum_{k=1}^{\infty} | \lt f,e_{k} \gt |^{2}= \|f\|^{2}\), \(\forall f \in H \)
- 5. If \( \lt f,e_{k} \gt =0\) then \( f = 0\)
Biorthogonal Bases
Definition: Sets of vectors \(\{\widetilde{e}_{k}\}_{k=1}^{\infty}\) and \(\{e_{k}\}_{k=1}^{\infty}\) constitute biorthogonal bases in \(H\) if
- 1. \( \forall k,j \in Z\), \( \lt e_{k},\widetilde{e}_{j} \gt =\delta_{k,j}\)
- 2. For any \(f \in H\) there exist \( A,B \gt 0\) such that \[ A\|f\|^{2} \le \sum_{k}| \lt f,e_{k} \gt |^{2} \le B\|f\|^{2} \] \[ \frac{1}{B}\|f\|^{2} \le \sum_{k}| \lt f,\widetilde{e}_{k} \gt |^{2} \le \frac{1}{A}\|f\|^{2} \]
Bases that satisfy the above two equations are called Riesz Bases. \( f \in H \) can be expanded as
\[ f=\sum_{k=1}^{\infty} \lt f,e_{k} \gt \widetilde{e}_{k} = \sum_{k=1}^{\infty} \lt f,\widetilde{e}_{k} \gt e_{k} \]Limitations of Bases or why we need Frames
Bases are characterized by their expansion property, ie. any function \(f \in H\) can be expressed as a linear combination of basis vectors \(e_{k}\) that span \(H\). However, it is possible to expand a function \(f \in H\) as a linear combination of another set of vectors \(\phi\) which may not be a basis in \(H\). The representation in this case may be redundant but it may provide additional flexibility which may be a good given specific applications. Another problem with basis is that they may be difficult to construct and a small change in implementing one vector in the basis may destroy the basis so having extra vectors in a redundant expansion may not be such a bad idea. In other applications, as in image compression, we may not want to keep all the coefficients of signal expansion with a set of given vectors so it is quite possible that frames may be able to do the job instead of bases which need more tedious construction.
Frames
Definition: Consider a set of vectors \(\{\phi_{k}\}_{k=1}^{\infty} \in H \) and that there exist constants \(A,B \gt 0\). \(\phi_{k}\) is a frame in \(H\) if
\[ A\|f\|^{2} \le \sum_{k=1}^{\infty} | \lt f,\phi_{k} \gt |^{2} \le B\|f\|^{2}, \forall f \in H \]If \(A=B\) then the frame is called a Tight Frame.
If vectors \(\{\phi_{k}\}_{k=1}^{\infty} \in H \) are linearly independent then the frame is non-redundant and is called a Riesz Basis.
Let \(\{\phi_{k}\}_{k=1}^{\infty} \in H \) be a frame with frame operator \(S\) and bounds \(A,B\) then following is true
- 1. S is bounded, invertible, adjoint and positive.
- 2. \(\{S^{-1}\phi_{k}\}_{k=1}^{\infty} \in H \) is a frame with frame operator \(S^{-1}\) and frame bounds \(A^{-1},B^{-1}\).
- 3. If \(A,B\) are optimal frame bounds for \(\{\phi_{k}\}_{k=1}^{\infty} \) then \(A^{-1},B^{-1}\) are optimal frame bounds for \(\{S^{-1}\phi_{k}\}_{k=1}^{\infty}\).
- 4. \( f=\sum_{k=1}^{\infty} \lt f,S^{-1}\phi_{k} \gt \phi_{k}, \forall f \in H \) and \( f=\sum_{k=1}^{\infty} \lt f,\phi_{k} \gt S^{-1}\phi_{k}, \forall f \in H \) Both converge for all \(f \in H\).
- 1. \( f=\sum_{k=1}^{\infty} \lt f,\chi_{k} \gt \phi_{k}, \forall f \in H \)
- 2. \( f=\sum_{k=1}^{\infty} \lt f,\phi_{k} \gt \chi_{k}, \forall f \in H \)
- 3. \( \lt f,g \gt =\sum_{k=1}^{\infty} \lt f,\phi_{k}\gt \lt \chi_{k},g \gt, \forall f,g \in H \)
Frame Operators and Pseudo-Inverse : Let \(U\) be the frame operator associated with the frame \(\{\phi_{k}\}_{k=1}^{\infty} \in H \) and let \(U: H \rightarrow H \) be bounded with closed range then \(\{U\phi_{k}\}_{k=1}^{\infty}\) is a frame sequence with frame bounds \(A\|U^{\dagger}\|^{-2}\) and \(B\|U\|^{2}\). \(U^{\dagger}\) is known as the pseudo inverse of the frame operator \(U\) and is defined as
\[ U^{\dagger}=(U^{*}U)^{-1}U^{*} \]where \(U^{*}\) is the adjoint of \(U\). As can be seen from the definition, \(U^{\dagger}\) is the left inverse.
Dual Frames: Assume that \(\{\phi_{k}\}_{k=1}^{\infty} \in H \) is an overcomplete frame then there exist frames \(\{\chi_{k}\}_{k=1}^{\infty} \in H \) for which \( f=\sum_{k=1}^{\infty} \lt f,\chi_{k} \gt \phi_{k}, \forall f \in H \) . Following holds for dual frames \(\phi_{k}\) and \(\chi_{k}\)