Prolate Spheroidal Wavefunctions and Wavelets:
The analytical theory of Multresolution Analysis (abbreviated MRA) is usually formulated in terms of a scaling function \phi which has the property that ${\phi(\cdot -k)}_{k \in \BB{Z}}$ forms a Riesz basis for a subspace $V_0$ of $L^2(\mathbb
)$. A nested chain of subspaces $V_n, n \in \mathbb
$ is then obtained by dilations of powers of two: $ f(\cdot) \in V_n iff f(\cdot*2^
) \in V_0$. The subspaces $V_n$ form a flag of $L^2(\mathbb
)$, ie:
[
V_n \subseteq V_
\quad n \in \mathbb
\bar
= L^2(\mathbb
)
\bigcap V_n = {0}
]
The construction of the MRA relies on the existence of a dilation equation, which expresses $\phi(\cdot * 2^{-1}) \in V_1$ as a linear combination of elements in $V_0$. The wavelets corresponding to the MRA are then created by modifying the dilation equation, and form bases for the subspaces $W_n = V_
- V_n$. As a result of this construction, wavelets have a strong time-frequency localization property that makes them well-suited to compressing signals arising from physical phenomena. However for the majority of wavelets the associated MRA is not stable under translation or differentiation. That is to say, if $\tau \in \mathbb
$ and $ f(\cdot) \in V_n$, then $f(\cdot - \tau)$ may fail to be in $V_n$ or indeed any $V_m$ for finite m. The same goes for diffferentiation. This problem makes the solution to pde's with periodic boundary conditions via Wavelet-Galerkin a difficult proposition. One notable exception is the Shannon MRA. This MRA decomposes $L^2(\mathbb
)$ into Paley-Weiner spaces PW_n of bandlimited functions with bandwidth $2^{-n}*2\pi$. The associated scaling function is the sinc function: $\phi
= sin(2\pi x)/2\pi x$. The Shannon MRA is translation and differentiation invariant, since $supp(2\pi \xi \hat
) = supp(2\pi \tau \hat
)= supp(\hat
)$. Despite these advantages, the Shannon MRA has often been dismissed as unsuitable for numerical computations due to the poor spacial localization of the sinc function. A natural remedy to the spacial localization problem is to try to find a basis for PW_n whose energy is maximally concentrated on some spacial interval. The solutions are given by the constrained optimization problem:
[\
\phi_
= \underset
\frac{\int_{-c}^
|\phi|2dx}{\int_{-\infty}
|\phi|^2dx}
]
The solutions $\phi_
$, known as prolate spheroidal wavefunctions (abbreviated PSWF's), were studied extensively by David Slepian and colleagues in a series of seminal papers (footnotes here, it would be nice if i could actually get my hands on all of these- theyre really hard to find. i talked to ncar library but havent heard back). PSWF's have a number of different characterizations that make them useful for time-frequency analysis.
(List various properties n good stuff here + talk about higher order PSWFs). While the PSWF's are entire functions, by choosing $c*n$ large enough they can be made arbitrarily small outside of the interval $(-c,c)$ and hence behave numerically like functions of compact support.
A change of scale in the above eqn leads to the relation:
[
\phi_
(2x) = frac
{\sqrt{2}} \phi_
]
This property was used in (Lilly, Walter et al ) to create a biorthogonal wavelet basis formally similar to the Shannon wavelet basis but with a scaling function that is suitable for numerical computations. This construction can be extended to periodic wavelets though i havent seen it anywhere in the literature. This is what we want to do.
Differentiation:
basically the point is that if $\phi = \phi_
$ is the PSWF scaling function, then $\phi^
\in V_0$ and has coefficients
[
\begin
a_n^
= {
\begin
\frac{k!(-1)^{n+k+1}}
&,& n \neq 0
\frac
&,& n = 0
\end
]
more generally if $f = \sum_n a_n \phi(x-n)$, then $f' \in V_0$ and:
[\
\begin
f' & = & \sum_n a_n \phi'(x-n)
& = & \sum_n a_n \sum_
\frac
\phi(x-n -k)
& = & \sum_j \sum_
\frac
a_n \phi(x -j)
\end
]
so youve got a hard bound on any error introduced by doing farge-style CVE stuff to speed up DNS (i dont think they have any truncation error bounds and seem to justify their methods by just using them and showing that for their particular eqns the error appears to be stable).
Translation:
here we have two analagous results. if $\phi = \phi_
$ is the PSWF scaling function, then $\phi(\cdot - \tau) \in V_0$ and has the expansion $\phi(x - \tau) = \sum_n sinc(n-\tau) \phi(x-n)$. When \tau is an integer the expansion coefficients $a_k$ reduce to $\delta_
$. For arbitrary $f \in V_0$ we have:
[\
\begin
f(x - \tau) & = & \sum_n a_n \phi(x-\tau-n)
& = & \sum_n a_n \sum_k sinc(k-\tau) \phi(x-n -k)
& = & \sum_j \sum_n sinc(j-n-\tau) a_n \phi(x -j)
\end
]
The point here is that since $\sum_k |sinc(x-k)|^2 \equiv 1 \quad \all x \in \mathbb
$ the amount of energy of a signal contained in V_k is invariant to translation (ie the wavelet coeffiecent tree is stable). Another way of saying this is that the operator given by orthog projection onto V_k commutes with the operator given by translation by $\tau$ for any $k \in mathbb{Z}. \tau \in \mathbb
$.
Biorthogonal wavelets are readily adaptable for use in Galerkin-Petrov schemes (cf- Wavelet theory and harmonic analysis in applied sciences- Carlos Enrique D'Attellis, Elena M. Fernández-Berdaguer p120 for a lit review).