CBMS: New Constructions of Functions Holomorphic in the Unit Ball of Cn (Regional Conference Series in Mathematics #63)

The starting point for the research presented in this book is A. B. Aleksandrov's proof that nonconstant inner functions exist in the unit ball $B$ of $C^n$. The construction of such functions has been simplified by using certain homogeneous polynomials discovered by Ryll and Wojtaszczyk; this yields solutions to a large number of problems. The lectures, presented at a CBMS Regional Conference held in 1985, are organized into a body of results discovered in the preceding four years in this field, simplifying some of the proofs and generalizing some results. The book also contains results that were obtained by Monique Hakina, Nessim Sibony, Erik Low and Paula Russo. Some of these are new even in one variable. An appreciation of techniques not previously used in the context of several complex variables will reward the reader who is reasonably familiar with holomorphic functions of one complex variable and with some functional analysis.

The book is based on lectures given by the author at a CBMS Regional Conference held in 1985. It presents the results about the construction of nonconstant inner functions in the unit ball in the $n$-dimensional complex space $C^n$. An appreciation of techniques not previously used in the context of several complex variables will reward the reader who is reasonably familiar with holomorphic functions of one complex variable and some functional analysis.

Uses as a starting point A B Aleksandrov's proof that nonconstant inner functions exist in the unit ball $B$ of $C^n$. This title simplifies the construction of such functions by using certain homogeneous polynomials discovered by Ryll and Wojtaszczyk; this yields solutions to a large number of problems.