By Jorge Mujica (Eds.)
This court cases quantity comprises papers of analysis of expository nature, and is addressed to analyze employees and complex graduate scholars in arithmetic. a number of the papers are the written and extended texts of lectures brought on the convention, while others were incorporated via invitation.
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Additional resources for Complex Analysis, Functional Analysis and Approximation Theory, Proceedings of the Conference on Complex Analysis and Approximation Theory Universidade Estadual de Campinas
And assume F is a Borel set in a B with C a p q ( E ) is a s e t of u n i q u e n e s s for H q if the r e s t r i c t i o n o p e r a t o r RE : ffq E q , which takes any function f € H q Let q E B > 0. We will say that E + to its generalized boundary value on for E, is injective. Our next result characterizes the boundary sets of uniqueness H q . The one dimensional case with q = - 1 is due to Hamilton 43 REPRODUCING KERNELS . (In this case H-l THEOREM 3 . 8 . b e a Bore1 s e t in is the classical space of Dirichlet finite functions in the unit disk).
1 1 p e r s i s t s and C j are allowed t o vary over t h e s e t D. E x E, zi , ,6 when 28 BEATROUS AN0 BURBEA We will present an abstract formulation of this result which admits a particularly simple proof based on Hilbert space considerations, and which yields a significant improvement of the classical result when specialized to the classical setting. For our purposes it will be more convenient to work with a slightly modified version of the Bergman-Schiffer inequality. Let K, and K 2 be positive definite kernels on X, and X 2 respectively.
Given by classical condition that E be a set of uniqueness is not required. In general some such assumption is essential. Korhyi and Pukgnski [141 gave an exampleof a function on a 2 point set in the bi-disk that is not the restricK tion of any function in the unit ball of Hm. For the kernels 1-I introduced above, it is clear that it suffices to assume that E is a set of uniqueness for the holomorphic subspace of L 1 fdul. Moreover, is a multiplier on if for each fixed 5 E D the function K( * , 5 1 HK, then it suffices to assume that E is a set of uniqueness for HK since in this case HK contains the class of functions used in the 36 BEATROUS AND BURBEA theorem.