Slepian function

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Slepian functions are a class of spatio-spectrally concentrated functions^[1] (that is, space- or time-concentrated while spectrally bandlimited, or spectral-band-concentrated while space- or time-limited) that form an orthogonal basis for bandlimited or spacelimited spaces.^[2][3][4][5] They are widely used as basis functions for approximation^[6] and in linear inverse problems^[7][8], and as apodization tapers or window functions^[9] in quadratic problems^[10] of spectral density estimation.^[11][12]

Slepian function constructions exist in discrete and continuous varieties, in one, two, and three dimensions, in Cartesian and spherical geometry, on surfaces and in volumes, on graphs^[13], and in scalar, vector,^[14] and tensor forms.^[15]

Without reference to any of these particularities^[16] , let ${\displaystyle f}$ be a square-integrable function of physical space, and let ${\displaystyle {\mathcal {H}}}$ represent Fourier transformation, such that ${\displaystyle F={\mathcal {H}}f}$ and ${\displaystyle {\mathcal {H}}^{-1}F=f}$. Let the operators ${\displaystyle {\mathcal {R}}}$ and ${\displaystyle {\mathcal {L}}}$ project onto the space of spacelimited functions, ${\displaystyle {\mathcal {S}}_{R}}$, and the space of bandlimited functions, ${\displaystyle {\mathcal {S}}_{L}}$, respectively, whereby ${\displaystyle R}$ is an arbitrary nontrivial subregion of all of physical space, and ${\displaystyle L}$ an arbitrary nontrivial subregion of spectral space. Thus, the operator ${\displaystyle {\mathcal {R}}}$ acts to spacelimit, and the operator ${\displaystyle {\mathcal {H}}^{-1}{\mathcal {L}}{\mathcal {H}}}$ acts to bandlimit the function ${\displaystyle f}$.

Slepian's quadratic spectral concentration problem aims to maximize the concentration of spectral power to a target region ${\displaystyle L}$, for a function that is spatially limited to a target region ${\displaystyle R}$. Conversely, Slepian's spatial concentration problem maximizes the spatial concentration to ${\displaystyle R}$ of a function bandlimited to ${\displaystyle L}$. Using ${\displaystyle \langle \cdot ,\cdot \rangle }$ for the inner product both in the space and the spectral domains, both problems are stated equivalently using Rayleigh quotients in the form

${\displaystyle \lambda ={\frac {\langle {\mathcal {R}}{\mathcal {H}}^{-1}{\mathcal {L}}\,F,{\mathcal {R}}{\mathcal {H}}^{-1}{\mathcal {L}}\,F\,\rangle }{\langle {\mathcal {H}}^{-1}{\mathcal {L}}\,F,{\mathcal {H}}^{-1}{\mathcal {L}}\,F\,\rangle }}={\frac {\langle {\mathcal {L}}{\mathcal {H}}{\mathcal {R}}f,{\mathcal {L}}{\mathcal {H}}{\mathcal {R}}f\rangle }{\langle {\mathcal {H}}{\mathcal {R}}f,{\mathcal {H}}{\mathcal {R}}f\rangle }}={\mbox{maximum}}.}$

The equivalent spectral-domain and spatial-domain eigenvalue equations are

${\displaystyle ({\mathcal {L}}{\mathcal {H}}{\mathcal {R}}{\mathcal {H}}^{-1\!}{\mathcal {L}})({\mathcal {L}}\,F\,)=\lambda ({\mathcal {L}}\,F\,)}$ and ${\displaystyle ({\mathcal {R}}{\mathcal {H}}^{-1\!}{\mathcal {L}}{\mathcal {H}}{\mathcal {R}})({\mathcal {R}}f)=\lambda ({\mathcal {R}}f),}$

given that ${\displaystyle {\mathcal {H}}}$ and ${\displaystyle {\mathcal {H}}^{-1}}$ are each others' adjoints, and that ${\displaystyle {\mathcal {R}}}$ and ${\displaystyle {\mathcal {L}}}$ are self-adjoint and idempotent.

The Slepian functions are solutions to either of these types of equations with positive-definite kernels, that is, they are bandlimited functions ${\displaystyle G={\mathcal {L}}F}$, concentrated to the spatial domain within ${\displaystyle R}$, or spacelimited functions of the form ${\displaystyle h={\mathcal {R}}f}$, concentrated to the spectral domain within ${\displaystyle L}$.

Let ${\displaystyle g(t)}$ and its Fourier transform ${\displaystyle G(\omega )}$ be strictly bandlimited in angular frequency between ${\displaystyle [-W,W]}$. Attempting to concentrate ${\displaystyle g(t)}$ in the time domain, to be contained within the time interval ${\displaystyle [-T,T]}$, amounts to maximizing

${\displaystyle \lambda ={\frac {\int _{-T}^{T}g^{2}(t)\,dt}{\int _{-\infty }^{\infty }g^{2}(t)\,dt}}={\text{maximum}},}$

which is equivalent to solving either, in the frequency domain, the convolutional integral eigenvalue (Fredholm) equation

${\displaystyle \int _{-W}^{W}D_{T}(\omega ,\omega ')G(\omega ')\,d\omega '=\lambda G(\omega ),\qquad D_{T}(\omega ,\omega ')={\frac {\sin T(\omega -\omega ')}{\pi (\omega -\omega ')}},\qquad |\omega |\leq W}$,

or the time- or space-domain version

${\displaystyle \int _{-T}^{T}D_{W}(t,t')g(t')\,dt'=\lambda g(t),\qquad D_{W}(t,t')={\frac {\sin W(t-t')}{\pi (t-t')}}=(2\pi )^{-1}\int _{-W}^{W}e^{i\omega (t-t')}d\omega ,\qquad t\in \mathbb {R} }$.

Either of these can be transformed and rescaled to the dimensionless

${\displaystyle \int _{-1}^{1}D(x,x')g(x')\,dx'=\lambda g(x),\qquad D(x,x')={\frac {\sin TW(x-x')}{\pi (x-x')}}}$.

The trace of the positive definite kernel is the sum of the infinite number of real and positive eigenvalues,

${\displaystyle N=\sum _{\alpha =1}^{\infty }\lambda _{\alpha }=\int _{-1}^{1}D(x,x')\,dx={\frac {2TW}{\pi }},}$

that is, the area of the concentration domain in time-frequency space (a time-bandwidth product).

One-dimensional scalar Slepian functions or tapers^[17] are the workhorse of the Thomson multitaper method of spectral density estimation.

We use ${\displaystyle g(\mathbf {x} )}$ and its Fourier transform ${\displaystyle G(\mathbf {k} )}$ to denote a function that is strictly bandlimited to ${\displaystyle {\mathcal {K}}}$, an arbitrary subregion of the spectral space of spatial wave vectors.^[18] Seeking to concentrate ${\displaystyle g(\mathbf {x} )}$ into a finite spatial region ${\displaystyle R\in \mathbb {R} ^{2}}$, of area ${\displaystyle A}$, we must find the unknown functions for which

${\displaystyle \lambda ={\frac {\int _{R}g^{2}(\mathbf {x} )\,d\mathbf {x} }{\int _{-\infty }^{\infty }g^{2}(\mathbf {x} )\,d\mathbf {x} }}={\mbox{maximum}}.}$

Maximizing this Rayleigh quotient requires solving the Fredholm integral equation

${\displaystyle \int _{\mathcal {K}}D_{R}(\mathbf {k} ,\mathbf {k} ')\,G(\mathbf {k} ')\,d\mathbf {k} '=\lambda G(\mathbf {k} ),\qquad D_{R}(\mathbf {k} ,\mathbf {k} ')=(2\pi )^{-2}\int _{R}e^{i(\mathbf {k} '-\mathbf {k} )\cdot \mathbf {x} }\,d\mathbf {x} ,\qquad \mathbf {k} \in {\mathcal {K}}.}$

The corresponding problem in the spatial domain is

${\displaystyle \int _{R}\!D_{\mathcal {K}}(\mathbf {x} ,\mathbf {x} ')\,g(\mathbf {x} ')\,d\mathbf {x} '=\lambda g(\mathbf {x} ),\qquad D_{\mathcal {K}}(\mathbf {x} ,\mathbf {x} ')=(2\pi )^{-2}\int _{\mathcal {K}}e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}\,d\mathbf {k} ,\qquad \mathbf {x} \in \mathbb {R} ^{2}.}$

Concentration to the disk-shaped spectral band ${\displaystyle {\mathcal {K}}=\{\mathbf {k} :\|\mathbf {k} \|\leq K\}}$ allows us to rewrite the spatial kernel as

${\displaystyle D_{\mathcal {K}}(\mathbf {x} ,\mathbf {x} ')={\frac {KJ_{1}(K\|\mathbf {x} -\mathbf {x} '\|)}{2\pi \|\mathbf {x} -\mathbf {x} '\|}},}$

with ${\displaystyle J_{1}}$ a Bessel function of the first kind, from which we may derive that

${\displaystyle N=\sum _{\alpha =1}^{\infty }\lambda _{\alpha }=\int _{R}D(\mathbf {x} ,\mathbf {x} ')\,d\mathbf {x} =K^{2}{\frac {A}{4\pi }},}$

in other words, again the area of the concentration domain in space-frequency space (a space-bandwidth product).

We denote ${\displaystyle g(\mathbf {\hat {r}} )}$ a function on the unit sphere ${\displaystyle \Omega }$ and its spherical harmonic transform coefficient ${\displaystyle g_{lm}}$ at the degree ${\displaystyle l}$ and order ${\displaystyle m}$, respectively,^[16] and we consider bandlimitation to spherical harmonic degree ${\displaystyle L}$, that is, ${\displaystyle g\in {\mathcal {S}}_{L}}$. Maximizing the quadratic energy ratio within the spatial subdomain ${\displaystyle R\subset \Omega }$ via

${\displaystyle \lambda ={\frac {\int _{R}g^{2}(\mathbf {\hat {r}} )\,d\Omega }{\int _{\Omega }g^{2}(\mathbf {\hat {r}} )\,d\Omega }}={\mbox{maximum}}}$

amounts in the spectral domain to solving the algebraic eigenvalue equation

${\displaystyle \sum _{l'=0}^{L}\sum _{m'=-l'}^{l'}D_{lm,l'm'}g_{l'm'}=\lambda g_{lm},\qquad D_{lm,l'm'}=\int _{R}Y_{lm}(\mathbf {\hat {r}} )Y_{l'm'}(\mathbf {\hat {r}} )\,d\Omega }$,

with ${\displaystyle Y_{lm}}$ the spherical harmonic at degree ${\displaystyle l}$ and order ${\displaystyle m}$. The equivalent spatial-domain equation, ${\displaystyle \int _{R}D(\mathbf {\hat {r}} ,\mathbf {\hat {r}} ')g(\mathbf {\hat {r}} )\,d\Omega ==\lambda g(\mathbf {\hat {r}} ),\qquad D(\mathbf {\hat {r}} ,\mathbf {\hat {r}} ')==\sum _{l=0}^{L}\sum _{m=-l}^{l}Y_{lm}(\mathbf {\hat {r}} )Y_{lm}(\mathbf {\hat {r}} ')=\sum _{l=0}^{L}\left({\frac {2l+1}{4\pi }}\right)P_{l}(\mathbf {\hat {r}} \cdot \mathbf {\hat {r}} '),}$ is a homogeneous Fredholm integral equation of the second kind, with a finite-rank, symmetric, separable kernel.

The last equality is a consequence of the spherical harmonic addition theorem which involves ${\displaystyle P_{l}}$, the Legendre polynomial. The trace of this kernel is given by

${\displaystyle N=\sum _{\alpha =1}^{(L+1)^{2}}\lambda _{\alpha }=\int _{R}D(\mathbf {\hat {r}} ,\mathbf {\hat {r}} )\,d\Omega =\sum _{l=0}^{L}\sum _{m=-l}^{l}D_{lm,lm}=(L+1)^{2}{\frac {A}{4\pi }}}$,

that is, once again a space-bandwidth product, of the dimension of ${\displaystyle {\mathcal {S}}_{L}}$ and the fractional area of ${\displaystyle R}$ on the unit sphere ${\displaystyle \Omega }$, namely ${\displaystyle A/(4\pi )}$.

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