Abstract
For the special case of a Riemann surface which arises as the double of a planar domainR, the trisecant identity has a natural interpretation as a relation among reproducing kernels for subspaces of the Hardy spaceH 2 (R). This relation and Riemann's theorem on the vanishing of the theta function is applied to Nevanlinna-Pick interpolation onR.
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References
M.B. Abrahmse,The pick interpolation theorem for finitely connected domains, Mich. Math.26 (1979), 195–203.
M.B. Abrahmse and R.G. Douglas,A class of subnormal operators related to a multiply connected domain. Adv. Math.121 (1976), 106–148.
J.A. Ball,A lifting theorem for operator models of finite rank on multiply connected domains, J. Operator Theory1 (1979), 3–25.
J.A. Ball and K. Clancey,Reproducing kernels for Hardy spaces on multiply connected domains, preprint.
S. Bell,The Szego projection and the classical objects of potential theory in the plane, Duke Math. J.64 (1991).
S. Bergman,The kernel function and conformal mapping, A.M.S. Surveys 5, A.M.S. Providence, 1950.
K. Clancey,Applications of the theory of theta functions to Hardy spaces of representing measures on multiply connected domains preprint.
J.D. Fay,Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, vol. 352, Springer-Verlag, 1973.
S. McCullough and L. C. Shen,On Szego's kernel of an annulus, Proceedings of the A.M.S. (to appear).
Mumford,Tata Lectures on Theta I, Birkhauser, 1983.
Mumford,Tata Lectures on Theta II, Birkhauser, 1983.
Z. Nehari,Conformal mapping, Dover Publications, Inc., New York, 1975.
D. Sarason,Generalized interpolation in H ∞, American Math society Transactions27 (1967), 180–203.
L.C. Shen,On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5 preprint.
Widom,Extremal polynomials associated with a system of curves in the complex plane, Advances in Math3 (1969), 127–232.