Let D be the interior of the
unit circle in C, Dc its exterior and T the unit circumference. We consider certain
piecewise holomorphic functions that are holomorphic in D and also in Dc. 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 P1. The set of all piecewise holomorphic functions (holomorphic in
D and in Dc) 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) measures; 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 P1 of P of representable
functions and M, as well as the ring structures in M (under convolution) and
P1 (Hadamard products), and questions of derivatives and integrals. We
end with an application to Fourier-Stieltjes moments relative to measues in
M.
