Vol. 72, No. 1, 1977

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On integral representations of piecewise holomorphic functions

G. K. Kalisch

Vol. 72 (1977), No. 1, 105–114

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.

Mathematical Subject Classification
Primary: 30A76, 30A76
Secondary: 30A86
Received: 1 November 1976
Revised: 23 March 1977
Published: 1 September 1977
G. K. Kalisch