On generalized prolate spheroidal functions

Abstract

Prolate spheroidal wave functions provide a natural and effective tool for computing with bandlimited functions defined on an interval. As demonstrated by Slepian et al., the so called generalized prolate spheroidal functions (GPSFs) extend this apparatus to higher dimensions. While the analytical and numerical apparatus in one dimension is fairly complete, the situation in higher dimensions is less satisfactory. This report attempts to improve the situation by providing analytical and numerical tools for GPSFs, including the efficient evaluation of eigenvalues, the construction of quadratures, interpolation formulae, etc. Our results are illustrated with several numerical examples.

On simulation of continuous determinantal point processes

“…If we let K c denote the Fourier transform of the unit ball, i.e. (see [19]) By choosing c " 2{α, we deduce that ψ is an eigenfunction of the Bessel-type kernel (20) associated with the eigenvalue ρΓ p1 `d{2q |γ| 2 {π d{2 . The generalized prolate spheroidal functions ψ and the eigenvalues γ have rather complicated expressions.…”

Quantum Limited Superresolution of Extended Sources in One and Two Dimensions

“…(27) For ν = φ, we may evaluate Eq. ( 27) by taking a product of expression (22) evaluated for µ = l and ∂ φ ρ1 , and then taking the diagonal matrix element of the product in state |λ i . A subsequent use of the second of the identities of Eq.…”

“…where and henceforth we omit any reference to the index set R as being understood without having to be explicitly stated. If we now set µ = φ in expression (22) and note that ∂ φ K xl |λ i , being merely the complex conjugate of the first identity in Eq. ( 26), vanishes, we may simplify expression (31),…”

“…( 50), we immediately recognize the coefficient functions C λ (r), of which there are countably infinitely many, as being the 2D version of the orthogonal set of generalized PSWFs first discussed by Slepian [6], with the quantity, c = 2πa, serving as the space-bandwidth parameter (SBP). Many of their properties were discovered by Slepian himself in that seminal paper, but a few subsequent papers [20][21][22] have derived further exact and asymptotic properties and provided useful analytical and numerical bases for them, while reviewing and simplifying Slepian's analysis. Here we will simply list some of those properties that are immediately relevant to our problem.…”

“…In the present paper, we have developed, as we show in Appendix B, an alternative approach to derive the associated eigenfunctions, the 2D radial PSWFs, Ψ N n (r), in their Bessel form and computed the associated eigenvalues λ N n by numerically diagonalizing an infinitedimensional matrix system of equations to which the eigenvalue equation ( 50) is equivalent along the radial dimension. In a rather tedious manner using certain integral identities involving the radial Zernike polynomials, Slepian [6,22] derived such a Bessel form for the 2D radial PSWFs, but our approach, in sharp contrast, is simpler, more constructive, and less tedious.…”