0.4 Bases, orthogonal bases, biorthogonal bases, frames, tight

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Development of ideas of vector expansion

Most people with technical backgrounds are familiar with the ideas of expansion vectors or basis vectors and of orthogonality; however, therelated concepts of biorthogonality or of frames and tight frames are less familiar but also important. In the study of wavelet systems, we find thatframes and tight frames are needed and should be understood, at least at a superficial level. One can find details in [link] , [link] , [link] , [link] , [link] . Another perhaps unfamiliar concept is that of an unconditional basis usedby Donoho, Daubechies, and others [link] , [link] , [link] to explain why wavelets are good for signal compression, detection, and denoising [link] , [link] . In this chapter, we will very briefly define and discuss these ideas. At this point, you may want to skip thesesections and perhaps refer to them later when they are specifically needed.

Bases, orthogonal bases, and biorthogonal bases

A set of vectors or functions ${f}_{k}\left(t\right)$ spans a vector space $F$ (or $F$ is the Span of the set) if any element of that space can be expressed as a linear combination of members of thatset, meaning: Given the finite or infinite set of functions ${f}_{k}\left(t\right)$ , we define ${\mathrm{Span}}_{k}\left\{{f}_{k}\right\}=F$ as the vector space with all elements of the space of the form

$g\left(t\right)=\sum _{k}\phantom{\rule{0.277778em}{0ex}}{a}_{k}\phantom{\rule{0.277778em}{0ex}}{f}_{k}\left(t\right)$

with $k\in \mathbf{Z}$ and $t,a\in \mathbf{R}$ . An inner product is usually defined for this space and is denoted $⟨f\left(t\right),g\left(t\right)⟩$ . A norm is defined and is denoted by $\parallel f\parallel =\sqrt{⟨f,f⟩}$ .

We say that the set ${f}_{k}\left(t\right)$ is a basis set or a basis for a given space $F$ if the set of $\left\{{a}_{k}\right\}$ in [link] are unique for any particular $g\left(t\right)\in F$ . The set is called an orthogonal basis if $⟨{f}_{k}\left(t\right),{f}_{\ell }\left(t\right)⟩=0$ for all $k\ne \ell$ . If we are in three dimensional Euclidean space, orthogonal basis vectors are coordinate vectors that are at right (90 o ) angles to each other. We say the set is an orthonormal basis if $⟨{f}_{k}\left(t\right),{f}_{\ell }\left(t\right)⟩=\delta \left(k-\ell \right)$ i.e. if, in addition to being orthogonal, the basis vectors are normalized to unity norm: $\parallel {f}_{k}\left(t\right)\parallel =1$ for all $k$ .

From these definitions it is clear that if we have an orthonormal basis, we can express any element in the vector space, $g\left(t\right)\in F$ , written as [link] by

$g\left(t\right)=\sum _{k}⟨g\left(t\right),\phantom{\rule{0.166667em}{0ex}}{f}_{k}\left(t\right)⟩\phantom{\rule{0.277778em}{0ex}}{f}_{k}\left(t\right)$

since by taking the inner product of ${f}_{k}\left(t\right)$ with both sides of [link] , we get

${a}_{k}=⟨g\left(t\right),\phantom{\rule{0.166667em}{0ex}}{f}_{k}\left(t\right)⟩$

where this inner product of the signal $g\left(t\right)$ with the basis vector ${f}_{k}\left(t\right)$ “picks out" the corresponding coefficient ${a}_{k}$ . This expansion formulation or representation is extremely valuable. It expresses [link] as an identity operator in the sense that the inner product operates on $g\left(t\right)$ to produce a set of coefficients that, when used to linearly combine the basis vectors, gives back the original signal $g\left(t\right)$ . It is the foundation of Parseval's theorem which says the norm or energycan be partitioned in terms of the expansion coefficients ${a}_{k}$ . It is why the interpretation, storage, transmission, approximation, compression, andmanipulation of the coefficients can be very useful. Indeed, [link] is the form of all Fourier type methods.

Although the advantages of an orthonormal basis are clear, there are cases where the basis system dictated by the problem is not and cannot (orshould not) be made orthogonal. For these cases, one can still have the expression of [link] and one similar to [link] by using a dual basis set ${\stackrel{˜}{f}}_{k}\left(t\right)$ whose elements are not orthogonal to each other, but to the corresponding element of the expansion set

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