CBMS Regional Conference Series in Mathematics 1986; 78 pp; softcover Number: 63 ISBN10: 0821807137 ISBN13: 9780821807132 List Price: US$32 Member Price: US$25.60 All Individuals: US$25.60 Order Code: CBMS/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 Løw 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. Readership Table of Contents  The pathology of inner functions
 \(RW\)sequences
 Approximation by \(E\)polynomials
 The existence of inner functions
 Radial limits and singular measures
 \(E\)functions in the Smirnov class
 Almost semicontinuous functions and \(\tilde{A}(B)\)
 \(u+vf\)
 Approximation in \(L^{1/2}\)
 The \(L^1\)modification theorem
 Approximation by inner functions
 The LSC property of \(H^\infty\)
 Maxsets and nonapproximation theorems
 Inner maps
 A Lusintype theorem for \(A(B)\)
 Continuity on open sets of full measure
 Composition with inner functions
 The closure of \(A(B)\) in \((LH)^p(B)\)
 Open problems
 Appendix I. Bounded bases in \(H^2(B)\)
 Appendix II. RWsequences revisited
 References
