A finite difference construction of the spheroidal wave functions

A fast and simple finite difference algorithm for computing the spheroidal wave functions is described. The resulting eigenvalues and eigenfunctions for real and complex spheroidal bandwidth parameter, c , agree with those in the literature from four to more than eleven significant figures. The validity of this algorithm in the extreme parameter regime, up to c² = 10¹⁴ , is demonstrated. Furthermore, the algorithm generates the spheroidal functions for complex order m . The coefficients of the differential equation can be simply modified so that the algorithm may solve any second order differential equation in Sturm–Liouville form. The prolate spheroidal functions and the spectral concentration problem in relation to band-limited and time-limited signals is discussed. We review the properties of these eigenfunctions in the context of Sturm–Liouville theory and the implications for a finite difference algorithm. A number of new suggestions for data fitting using prolate spheroidal wave functions with a heuristic for optimally choosing the value of c and the number of basis functions are described.