Finite interpolation for analytic functions with finite Dirichlet integrals

Abstract

PROOF. The necessity of the condition R (IOAD is evident. We have to show the solvability of (1) under the condition R GOAD. Since R has finite genus, R GOAD implies the existence of a nonconstant ABDfunction F(z) on R. Let R* be a closed Riemann surface which contains R as a subsurface. Choose a point Po in R{1, 12, . . . , rnI such that F(to) 5 F(?k) (k = 1, 2, . .. , n). For each fixed k (k = 1, 2, ... , n), by Riemann-Roch's theorem, there exists a meromorphic function rk(z) on R* such that rk(z) has a simple pole at rk and a pole of order nk at to and regular on R*-{ -o, 0k }. Let mk be the order of zero of the function 1flL (F(z) F(?j))m+l at rk and let s=max {mknk; k=1, 2, * ,n}. Put