Geometric conditions for the reconstruction of a holomorphic function by an interpolation formula

TL;DR

This paper investigates geometric conditions that ensure the accurate reconstruction of holomorphic functions in complex two-dimensional space from their restrictions on complex lines, with implications for economics and medical imaging.

Contribution

It provides geometric criteria and stability conditions for the explicit reconstruction of holomorphic functions from line restrictions in a7a7 2D complex space.

Findings

Reconstruction validity depends on the mutual distribution of lines.

A geometric description of the stability condition is provided.

Stronger stability is linked to permutations and subfamilies of lines.

Abstract

We give here some precisions and improvements about the validity of the explicit reconstruction of any holomorphic function on a ball of $\mathbb{C}^2$ from its restrictions on a family of complex lines. Such validity depends on the mutual distribution of the lines. This condition can be geometrically described and is equivalent to a stronger stability of the reconstruction formula in terms of permutations and subfamilies of these lines. The motivation of this problem comes from possible applications in mathematical economics and medical imaging.