Generalized prolate spheroidal functions (GPSFs) subroutines

This repository includes Fortran codes for computing with generalized prolate spheroidal functions (GPSFs), which are the extension of prolate spheroidal wave functions to greater than one dimension. These codes are an implementation of the algorithms of https://arxiv.org/abs/1811.02733 and include subroutines for evaluating GPSFs, quadrature of bandlimited functions, and approximation via GPSF expansion.

Codes for computing with GPSFs are included in gpsfexps.f and were written by Philip Greengard and Kirill Serkh. All *.f files other than gpsfexps.f and xk_exps.f were written by Vladimir Rokhlin who generously provided them for use in this repository.

As of 10/9/2022, these codes have not been seriously optimized. There are several potential speedups that could be made including use of fast numerical linear algebra software.

These codes were originally written and tested in a Linux environment using the "f90" compiler and have also been tested on several Macs using the gfortran compiler.

Numerical tests of equations in the paper

In gpsfexps.f we include tests of a few equations in the paper. We reference formulas using the original arxiv version of the paper https://arxiv.org/abs/1811.02733.

• Evaluating GPSFs: the subroutine plot_phi_88 plots the radial component of GPSFs defined in (87) and (88). Plotting is done using gnuplot.
• Eigenfunctions of linear operator $L_{N, c}$: subroutine test_lnc_93 checks equation (93)
• Eigenfunctions of integral operator $M_{N, c}$: subroutine test_mnc_90 checks equation (90)
• Evaluation of eigenvalues of GPSF integral operator: subroutine test_gammas_4 checks the algorithms of Sections 4.1 and 4.2
• Second-order ODE satisfied by GPSFs: subroutine test_ode_236 checks equation (236)