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New Constructions of Functions Holomorphic in the Unit Ball of $$C^n$$
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1986; 78 pp; softcover
Number: 63
ISBN-10: 0-8218-0713-7
ISBN-13: 978-0-8218-0713-2
List Price: US$30 Member Price: US$24
All Individuals: US\$24
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.

• 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 $$H^\infty$$
• Max-sets and nonapproximation theorems
• Inner maps
• A Lusin-type 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. RW-sequences revisited
• References