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Journal of the London Mathematical Society 1982 s2-25(2):297-304; doi:10.1112/jlms/s2-25.2.297
© 1982 by London Mathematical Society
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© Oxford University Press

Minimal Interpolation for Harmonic Functions

Eliyahu Beller, Stephen D. Fisher and Bernard Pinchuk

Department of Mathematics, Bar-Ilan University Ramat-Gan, Israel
Department of Mathematics, Northwestern University Evanston, Illinois 60201, U.S.A.
Technion, Israel Institute of Technology Haifa, Israel

Let K be a closed convex set in Cn and let z1,...,zn be distinct points in the open unit disc of the complex plane, with no zj = 0. A description is given of those functions f of minimal h1 norm which satisfy the interpolation conditions (f(z1),...,f(zn)){varepsilon}K. The unique extremal is found in the case when n = 1. The situation when one of the interpolation points is the origin is analyzed as well. A slightly more general problem is examined in hp for 1 < p ≤ {infty} the results here are more routine and are included for completeness.


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