Geometric conditions for the reconstruction of a holomorphic function by an interpolation formula. (English) Zbl 1370.32004

Irigoyen, Amadeo

Dolomites Res. Notes Approx. 9, 1-15 (2016).

Summary: 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.

MSC:

32E30	Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
30E05	Moment problems and interpolation problems in the complex plane
41A05	Interpolation in approximation theory
94A08	Image processing (compression, reconstruction, etc.) in information and communication theory