A construction of outer functions along curves on the sphere in ℂⁿ

For a simple smooth curv Γe F on the unit sphere in ℂn transverse at all its points, and a non-negative continuous function φ on Γ satisfying ∫Γ log φp ds > - ∞ together with a finite number of necessary compatibility conditions, an holomorphic function f on the unit ball is constructed such that |f|= φ on Γ. The case of a general, not necessarily transverse, curve is also investigated. © 1996 Academic Press, Inc.