The 1-dimensional extension property in complex analysis

A classical theorem states that if a function on the unit circle has vanishing negative Fourier coefficients, then it extends to holomorphic function on the unit disc. What happens when you are given a family of curves, and a function which extends holomorphically from each of the curves? This area of study is called the ”1-dimensional extension problem”. Results for planar domains, and for holomorphic extension from boundaries in C^n will be discussed. One application is the construction of a completely new class of algebras of real analytic functions. Various techniques of analytic extension in several variables are used to prove these results.