Slepian function
Slepian functions are a class of spatio-spectrally concentrated functions that form an orthogonal basis for bandlimited or spacelimited spaces. That is, they are concentrated in space or time while spectrally bandlimited, or concentrated in spectral band while space- or time-limited. They are widely used as basis functions for constructive approximation and in linear inverse problems, and as apodization tapers or window functions in quadratic problems of spectral density estimation.
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, and in scalar, vector, and tensor forms.
General setting and operator formalism
Without reference to any of these particularities, let be a square-integrable function of physical space, and let represent Fourier transformation, such that and.Let the operators and project onto the space of spacelimited functions,
, and the space of bandlimited functions,, respectively, whereby is an arbitrary nontrivial subregion of all of physical space, and an arbitrary nontrivial subregion of spectral space. Thus, the operator acts to spacelimit, and the operator
acts to bandlimit the function.
Slepian's quadratic spectral concentration problem aims to maximize the concentration of spectral power to a target region, for a function that is spatially limited to a target region . Conversely, Slepian's spatial concentration problem maximizes the spatial concentration to of a function bandlimited to. Using for the inner product both in the space and the spectral domains, both problems are stated equivalently using Rayleigh quotients in the form
The equivalent spectral-domain and spatial-domain eigenvalue equations are
and
given that and are each others' adjoints, and that
and 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
, concentrated to the spatial domain within, or spacelimited functions of the form
, concentrated to the spectral domain within.
Scalar Slepian functions in one dimension
Let and its Fourier transform be strictly bandlimited in angular frequency between. Attempting to concentrate in the time domain, to be contained within the time interval, amounts to maximizingwhich is equivalent to solving either, in the frequency domain, the convolutional integral eigenvalue equation
or the time- or space-domain version
Either of these can be transformed and rescaled to the dimensionless
The trace of the positive definite kernel is the sum of the infinite number of real and positive eigenvalues,
that is, the area of the concentration domain in time-frequency space.
One-dimensional scalar Slepian functions or tapers are the workhorse of the Thomson multitaper method of spectral density estimation.
Scalar Slepian functions in two Cartesian dimensions
We use and its Fourier transform to denote a function that is strictly bandlimited to, an arbitrary subregion of the spectral space of spatial wave vectors.Seeking to concentrate into a finite spatial region,
of area, we must find the unknown functions for which
Maximizing this Rayleigh quotient requires solving the Fredholm integral equation
The corresponding problem in the spatial domain is
Concentration to the disk-shaped spectral band
allows us to rewrite the spatial kernel as
with a Bessel function of the first kind, from which we may derive that
in other words, again the area of the concentration domain in space-frequency space.
Scalar Slepian functions on the surface of a sphere
We denote a function on the unit sphere and its spherical harmonic transform coefficient at the degreeand order, respectively, and we consider bandlimitation to spherical harmonic degree, that is,. Maximizing the quadratic energy ratio within the spatial subdomain via
amounts in the spectral domain to solving the algebraic eigenvalue equation
with the spherical harmonic at degree and order. The equivalent spatial-domain equation,
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, the Legendre polynomial. The trace of this kernel is given by
that is, once again a space-bandwidth product, of the dimension of and the fractional area of on the unit sphere, namely.