Draft:Slepian function

Slepian functions are a class of spatio-spectrally concentrated functions (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. They are widely used as basis functions for approximation and in linear inverse problems, and as apodization tapers or  window functions in quadratic problems of spectral density estimation.

Slepian functions exist in discrete and continuous varieties, in one, two, and three dimensions, in Cartesian and spherical geometry, on surfaces and in volumes, and in scalar, vector, and tensor forms.

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

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

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

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

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

given that $$\mathcal{H}$$ and $$\mathcal{H}^{-1}$$ are each others' adjoints, and that $$\mathcal{R}$$ and $$\mathcal{L}$$ 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 $$G=\mathcal{L}F$$, concentrated to the spatial domain within R, or spacelimited functions of the form $$h=\mathcal{R}f$$, concentrated to the spectral domain within L.

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

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

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

$$\int_{-W}^{W}D_T(\omega,\omega')G(\omega')\,d\omega'=\lambda G(\omega),\qquad D_T(\omega,\omega')=\frac{\sin T(\omega-\omega')}{\pi(\omega-\omega')}, \qquad |\omega|\le W$$,

or the time- or space-domain version

$$\int_{-T}^{T}D_W(t,t')g(t')\,dt'=\lambda g(t),\qquad D_W(t,t')=\frac{\sin W(t-t')}{\pi(t-t')}, \qquad t \in \mathbb{R}$$.

Either of these can be transformed and rescaled to the dimensionless

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

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

$$N=\sum_{\alpha=1}^\infty \lambda_\alpha=\int_{-1}^{1}D(x,x')\,dx=\frac{2TW}{\pi},$$

that is, the area of the concentration domain in time-frequency space.

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

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

Maximize this Rayleigh quotient requires solving the Fredholm integral equation

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

The corresponding problem in the spatial domain is

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