An approximation formula for holomorphic functions by interpolation on the ball

TL;DR

This paper presents an explicit interpolation formula for reconstructing holomorphic functions on the unit ball in c2 from their restrictions on complex lines, with convergence as the number of lines increases, motivated by applications in economics and medical imaging.

Contribution

It introduces a new explicit Lagrange-type interpolation formula for holomorphic functions based on line restrictions, with proven approximation properties.

Findings

Interpolation formula converges as the number of lines increases

Explicit construction from function values and derivatives on lines

Potential applications in economics and medical imaging

Abstract

We deal with a problem of the reconstruction of any holomorphic function $f$ on the unit ball of $\mathbb{C}^2$ from its restricions on a union of complex lines. We give an explicit formula of Lagrange interpolation's type that is constructed from the knowledge of $f$ and its derivatives on these lines. We prove that this formula approximates any function when the number of lines increases. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.