Total Positivity and Its Applications, M.Gasca and C. A. Micchelli, Eds.

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 115 KB
• Volume: 90
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1997.3150

✦ Synopsis

Part 2 contains five chapters, the first four of which are in parallel with the four chapters of part 1 (even with the same titles), but in the multivariate setting. The fifth chapter in part 2 deals with summability for the two-variable case using a method of Marcinkiewicz type. Several aspects of the theory of multiple trigonometric series, the existence and properties of the conjugates and Hilbert transforms, convergence (pointwise and in L p ), and summability are covered, very much in parallel with the development of the theory for one variable in part 1. However, special emphasis is placed on the new effects which arise when dealing with multiple series and which are not present in the one-variable case. Many results of the author are presented here, often with new proofs that were not published before.

All chapters follow the same pattern: first a brief survey of known results is given, then some deeper results are treated in more detail, and at the end of each chapter some unsolved problems are formulated. The latter makes this book a very useful source of research problems, recommended for new researchers in this area.