Preface to the Second Edition

Summary

In this second edition, several major changes have been made to the structure of the book. A new chapter on orthogonal polynomials in two variables has been added to provide a more convenient source of information for readers concerned with this topic. The chapter collects results previously scattered in the book, specializing results in several variables to two variables whenever necessary, and incorporates further results not covered in the first edition. We have also added a new chapter on orthogonal polynomials on the unit sphere, which consolidates relevant results in the first edition and adds further results on the topic. Since the publication of the first edition in 2001, considerable progress has been made in this research area. We have incorporated several new developments, updated the references and, accordingly, edited the notes at the ends of relevant chapters. In particular, Chapter 5, “Examples of Orthogonal Polynomials in Several Variables”, has been completely rewritten and substantially expanded. New materials have also been added to several other chapters. An index of symbols is given at the end of the book.

Another change worth mentioning is that orthogonal polynomials have been renormalized. Some families of orthogonal polynomials in several variables have expressions in terms of classical orthogonal polynomials in one variable. To provide neater expressions without constants in square roots they are now given in the form of orthogonal rather than orthonormal polynomials as in the first edition. The L² norms have been recomputed accordingly.