By Jorge Mujica (Eds.)

This court cases quantity comprises papers of study of expository nature, and is addressed to analyze staff and complicated graduate scholars in arithmetic. the various papers are the written and increased texts of lectures added 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**

**Sample text**

Thus it is natural to consider the problem of characterizing the functions on restrictions of bounded holomokphic functions on B . Let K be a positive definite kernel on a set E which are X, and let E be an arbitrary subset of X. We will say that a function f on E satisfies the P i c k - N e v a n l i n n a c o n d i t i o n on E (with respect to the kernel K ) if the kernel K l x , y ) l l - f l x ) f ( y l l is positive definite on E x E . 7) to the present setting we obtain: THEOREM 2 . 1 . Let K be a p o s i t i v e d e f i n i t e s e s q u i - h o l o m o r p h i c kernel H on a domain D such t h a t t h e a s s o c i a t e d f u n c t i o n a l H i l b e r t space contains the constants.

Conn 2 3 versely, it can be shown that every positive definite kernel is the .. reproducing kernel for some functional Hilbert space. 1. be a p o s i t i v e d e f i n i t e k e r n e l o n K X on There i s a unique f u n c t i o n a l H i l b e r t space (i) Then X. X with X as i t s reproducing kernel; I( Ho (ii) The S p a c e 8 cli K f * ,xl) i s d e n s e i n C ff; on X (iii) A f u n c t i o n f negative constant is in (iV) H i f and o n l y i f 2 C K(x,y) - t h w e i s anon__ such t h a t the kernel p o s i t i v e d e f i n i t e , and i n t h i s c a s e , of a l l s u c h f= c o n s i s t i n g o f f i n i t e sums o f t h e f o r m t h e norm o f f fix) ffy) i s the is infimum C; I f i s a n y s u b s e t of E with reproducing kernel KIE x F X is t h e n t k e f u n c t i o n a l Hilbert space fflE.

In this case, the z E D. 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.