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.
