This is volume 155 of the series Encyclopedia of Mathematics and its Applications. The series has books on topics with wide applicability and for which the details of the abstract theory is subordinate to the applications.
The present book is a thoroughly revised and updated version of the first edition (2001, vol. 81 of this series). It is at the same time an introduction to and a reference work on the topic of multivariate polynomials.
To move from one to several variables the problem is not only requiring a more complex analysis, because scalars become n-tuples. An essential difference in the case of orthogonal polynomials is that in the univariate case the degree of a polynomial is just one number which essentially allows just one possible ordering of the orthogonal sequence, while in the multivariate case the degree is an n-tuple and there are many ways to order these, so that also the sequence of orthogonal polynomials is not unique.
Those who are familiar with the fist edition will recognize of course its content, but they should be pointed to the two new chapters 2 and 3, and a considerable update of chapter 5 with explicit examples of orthogonal polynomials in several variables. Another feature is that the presentation is simplified, avoiding several constants that previously guaranteed orthonormality. Now the polynomials are just assumed to be orthogonal with a more convenient normalization.
Here is a short description of the contents.
- Background: Introduction to hypergeometric series and (classical) orthogonal polynomials in one variable.
- Orthogonal polynomials in two variables: This is a new chapter. It collects results that specifically hold for the two-variable case. Some are new, some are specializations of the general case and/or that were scattered throughout the first edition. This includes product type polynomials, orthogonal polynomials on the disc and the triangle, and the first and second Koornwinder families (orthogonal on triangles with curved boundaries).
- General properties of orthogonal polynomials in several variables: This generalizes the classical theory in one variable: moment problem, matrix form of the three-term recurrence relation, reproducing kernels, Fourier series, location of the zeros, and Gaussian cubature.
- Orthogonal polynomials on the unit sphere: Also this is a new chapter in the second edition. This is a multivariate generalization of orthogonal polynomials on the unit circle, which form an important class of univariate polynomials. However, the analysis is quite different. The simplest case are the spherical harmonics (orthogonal with respect to the surface measure, but a more general treatment follows in chapter 7). The general orthogonality is connected with orthogonality on the unit ball and on the simplex. This is a relatively brief chapter but it is a precursor preparing elements for several of the later chapters.
- Examples of orthogonal polynomials in several variables: General polynomial systems orthogonal with respect to separable product weights, or rotation invariant weights, orthogonal systems on the whole space (Hermite), the positive cone (Laguerre), the ball or the simplex. Furthermore Chebyshev polynomials generalizing Koornwinder's second family, and Sobolev orthogonality on the unit ball.
- Root systems and Coxeter groups: Finite Coxeter groups are finite reflection groups (like the symmetry groups of regular polyhedra). The relation with polynomials is that the ring of its invariants is isomorphic to the set of multivariate polynomials. Besides an introduction to these groups, the main topics are difference-differential operators (to understand non-invariant polynomials for invariant measures), intertwining operators (fractional integral transforms mapping a set of orthogonal polynomials into another one), and κ-analogs of the exponential (exponentials are of course essential in Fourier analysis).
- Spherical harmonics associated with reflection groups: This is about orthogonal polynomials on spheres for weights invariant under reflection groups so that we see the theory of the previous chapter in action. The homogeneous polynomials are developed along the same lines as the usual spherical harmonics. Orthogonality does hold for different inner products. Familiar results are generalized like reproducing kernels and Fourier series (Dunkl transform). Several concrete examples of orthogonal systems are given.
- Generalized classical orthogonal polynomials: With this chapter, one is even more on familiar ground. These are multivariate orthogonal polynomials that depend on a set of parameters and if some of these go to zero, then the usual univariate polynomials are recovered. The polynomials considered are the ones orthogonal on the unit ball and the simplex, and the generalizations of Hermite and Laguerre polynomials.
- Summability of orthogonal expansions: This is obviously about convergence of general orthogonal expansions (Fourier series). After a general result, convergence of partial sums, Cesàro means, and expansions in terms of homogeneous harmonics and the orthogonal polynomials from the previous chapters is discussed.
- Orthogonal polynomials associated with symmetric groups: Here the differential-difference operators become important instruments. The treatment is algebraic involving linear algebra and combinatorics. It results in explicit formulas for the case of particular weights on the torus and for the Gaussian weight on the whole space.
- Orthogonal polynomials associated with octahedral groups, and applications: This is the symmetric group but now including additionally signs, giving the symmetry groups for hypercubes and hyperoctahedra. Much is taken from the previous chapter requiring that they are even or odd in some of the variables. Applications can be found in quantum mechanics.
The book fits perfectly in the guidelines set out for this Encyclopedia series. Although the material is very theoretical and abstract, everything is kept concise, and wherever possible, concrete examples are worked out, and for example in the last chapter even an algorithm is described to compute the binomial coefficients for the nonsymmetric Jack polynomials. Also the applications are mentioned where possible. It's a definition-theorem-proof type of book. It introduces all the necessary material, and it follows the main flow of the ideas, with further technical and historical details banned to the end-notes for each chapter that forward the reader to the necessary literature.
Like in the first edition, some topics were not included: orthogonal polynomial related to representation theory of Lie groups, of q-type, quantum groups, and Macdonald symmetric polynomials. That has not changed with this second edition, but as described above, the second edition is a substantial update of the first.
The subject index and the index of symbols are helpful but they could have been more complete though. But let this be a minor critique of an otherwise truly encyclopedic piece of work on this particular topic that is not matched by any other book.