Vol. 15, No. 4, 1965

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Boundary measures of analytic differentials and uniform approximation on a Riemann surface

Laura Ketchum Kodama

Vol. 15 (1965), No. 4, 1261–1277

A classical theorem of F. and M. Riesz establishes a one-to-one correspondence between analytic differentials of class H1 on the interior of the unit disc and finite complex-valued Borel measures on the boundary of the disc which are orthogonal to polynomials. The main result of this paper gives a similar correspondence when the unit disc is replaced by a compact subset, satisfying a finite connectivity condition, of any noncompact Riemann surface. The analytic differentials on the interior of the set satisfy a boundedness condition analogous to the classical H1 differentials and the measures on the boundary of the set are those orthogonal to all meromorphic functions with a finite number of poles in the complement of the set. This result is then used to obtain theorems on uniform approximation on the set by such meromorphic functions.

Mathematical Subject Classification
Primary: 30.45
Received: 10 January 1964
Published: 1 December 1965
Laura Ketchum Kodama