A Discretized Fourier Orthogonal Expansion in Orthogonal Polynomials on a Cylinder

TL;DR

This paper analyzes a discretized Fourier orthogonal expansion on a cylindrical domain, providing convergence results and an algorithm for 3D image reconstruction in computed tomography.

Contribution

It introduces a discretized expansion using Radon projections and establishes convergence and Lebesgue constant bounds for functions on a cylinder.

Findings

Lebesgue constant grows as m (log(m+1))^2

Convergence proven for functions in C^2(B^2 x [-1,1])

Algorithm for 3D image reconstruction in tomography

Abstract

We study the convergence of a discretized Fourier orthogonal expansion in orthogonal polynomials on $B^2 \times [-1,1]$, where $B^2$ is the closed unit disk in $\RR^2$. The discretized expansion uses a finite set of Radon projections and provides an algorithm for reconstructing three dimensional images in computed tomography. The Lebesgue constant is shown to be $m \, (\log(m+1))^2$, and convergence is established for functions in $C^2(B^2 \times [-1,1])$.