Reconstruction of density functions by sk-splines

TL;DR

This paper develops a method for reconstructing density functions and their characteristic functions using radial basis functions on scattered data, addressing technical challenges posed by their non-compact domains.

Contribution

It provides explicit formulas for Fourier transforms of kernel functions and solves the interpolation problem on an infinite grid for density reconstruction.

Findings

Explicit Fourier transform formulas for kernel functions

Successful interpolation on infinite rectangular grids

Enhanced methods for density function reconstruction

Abstract

Reconstruction of density functions and their characteristic functions by radial basis functions with scattered data points is a popular topic in the theory of pricing of basket options. Such functions are usually entire or admit an analytic extension into an appropriate tube and "bell-shaped" with rapidly decaying tails. Unfortunately, the domain of such functions is not compact which creates various technical difficulties. We solve interpolation problem on an infinite rectangular grid for a wide range of kernel functions and calculate explicitly their Fourier transform to obtain representations for the respective density functions.