Reconstructing holomorphic functions in a domain from their values on a part of its boundary

This survey article is about the class of holomorphic functions that are representable by integral Carleman formulas. Unlike Cauchy's formula, these integral formulas represent a function holomorphic in a domain D in terms of its values on a subset M of the boundary partial derivative D, if M has a positive Lebesgue measure satisfying 0 < lambda(M) < lambda(partial derivative D). It is known that if f is an element of E-1(D), [GRAPHICS] where the quenching function phi is an element of E-infinity(D) satisfies 1) vertical bar phi(z)vertical bar = 1 a.e. on partial derivative D\M, 2) vertical bar phi(z)vertical bar > 1 for every z is an element of D. The problem, originally posed by L.A. Aizenberg, is the converse one: Suppose that f is holomorphic in D and has angular boundary values denoted also by f on M, such that f is an element of L-1 (M) and assume that f satisfies the above formula (where the convergence is assumed to be just point-wise). What can be said about f? In other words, the problem is to describe the class of functions to which f belongs (this space is larger than the space E-1(D)). We have found the answer for a large class of examples in the case of one complex variable when the set M is an Ahlfors regular curve. Nothing is known if M is just a set of positive measure. Here, to illustrate the main ideas, we present a new reconstruction formula (Carleman formula), based on ideas of Aizenberg, for a holomorphic function f in the interior of a convex polygon from the values of f on one of its sides. In the case of several complex variables, for similarly formulated problems (but without the use of quenching functions), we know even less. Some known cases require quite explicit computations, obscuring a deeper understanding of the problem.