NEHARI AND NEVANLINNA-PICK PROBLEMS AND HOLOMORPHIC EXTENSIONS IN THEPOLYDISK IN TERMS OF RESTRICTED BMO

A new class BMOr is introduced, motivated by the following extension o f the Nehari theorem to d greater-than-or-equal-to 1: The norm of a bo unded (big) Hankel form B in the d-dimensional torus satisfies \\phi\\ less-than-or-equal-to \\B\\ less-than-or-equal-to square-root d \\phi \\ where the norm of phi, the antianalytic symbol of B, is taken in BM Or. This extension in turn provides solutions to Nevanlinna-Pick type interpolation problems in the polydisk, with control of the BMOr norm. Necessary and sufficient conditions for the existence of H-2 and BMOr extensions of analytic functions defined on a zero variety V are give n, with control of the norm, under general conditions on V. (C) 1994 A cademic Press, Inc.