Processing math: 17%

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 $f \in H$.



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 $\chi_{k}$

• 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. $\lt f,g \gt =\sum_{k=1}^{\infty} \lt f,\phi_{k}\gt \lt \chi_{k},g \gt, \forall f,g \in H$

Hilbert Space

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Functional Analysis Wikibook

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Basis Wiki Page

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Frames Wiki page

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Ole Christensen::An introduction to frames and Riesz bases

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Mallat::A Wavelet Tour of Signal processing

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Goyal et al::Fourier and Wavelet Signal Processing

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Christopher Heil::A Basis Theory primer

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