# New Constructions of Functions Holomorphic in the Unit Ball of \(C^{n}\)

Share this pageA co-publication of the AMS and CBMS

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

# Table of Contents

## New Constructions of Functions Holomorphic in the Unit Ball of $C^{n}$

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Acknowledgments vii8 free
- Introduction ix10 free
- Notation xv16 free
- 1. The Pathology of Inner Functions 118 free
- 2. RW-Sequences 421
- 3. Approximation by E-Polynomials 825
- 4. The Existence of Inner Functions 1330
- 5. Radial Limits and Singular Measures 1633
- 6. E-Functions in the Smirnov Class 1835
- 7. Almost Semicontinuous Functions and Ã(B) 2138
- 8. |u + vf| 2340
- 9. Approximation in L[sup(1/2)] 2744
- 10. The L[sup(1)]-Modification Theorem 3047
- 11. Approximation by Inner Functions 3653
- 12. The LSC Property of H[sup(∞)] 3855
- 13. Max-Sets and Nonapproximation Theorems 4057
- 14. Inner Maps 4562
- 15. A Lusin-Type Theorem for A(B) 4966
- 16. Continuity on Open Sets of Full Measure 5572
- 17. Composition with Inner Functions 5976
- 18. The Closure of A(B) in (LH)[sup(p)](B) 6481
- 19. Open Problems 6683
- Appendix I. Bounded Bases in H[sup(2)](B) 7087
- Appendix II. RW-Sequences Revisited 7289
- References 7592
- Back Cover Back Cover196