On integral representations of piecewise holomorphic functions

Let D be the interior of the unit circle in C, D c its exterior and T the unit circumference. We consider certain piecewise holomorphic functions that are holomorphic in D and also in D c . This paper deals with those piecewise holomorphic functions that are representable by means of complex Poisson-Stieltjes integrals on T; we call this set of functions Pi. The set of all piecewise holomorphic functions (holomorphic in D and in D c ) we call P. Earlier work-see Rolf Nevanlinna, Eindeutige Analytische Funktionen, Springer, Berlin, 1953 and references there-dealt with positive (Herglotz-Riesz) or real (Nevanlinna) measres; we shall use here the entire space M of bounded complex Borel measures on T. This gives the theory more flexibility. We consider characterizations of functions in P representable by means of complex Poisson-Stieltjes integrals, uniqueness questions, the nature of the mapping between the subset P 1 of P of representable functions and M, as well as the ring structures in M (under convolution) and P 1 (Hadamard products), and questions of derivatives and integrals. We end with an application to Fourier-Stieltjes moments relative to measues in M.