Asymptotics for orthogonal polynomials
It is indeed a long time ago - as the author states in the preface - that a textbook on general orthogonal polynomials (on the real line) appeared. Aport from the by now classical books by \textit{G. Szegö} [Orthogonal Polynomials, Am. Math. Soc. Coll. Publ. 23 (1939; Zbl 0023.21505)] and \textit{G. Freud} [Orthogonal Polynomials (1966; Zbl 0169.080)] there are very few new contributions in textbook form. Of course one has the book by \textit{T. S. Chihara} [An introduction to orthogonal polynomials (1978; Zbl 0389.33008)] and the monographs by \textit{Ya. L. Geronimus} [Polynomials orthogonal on a circle and an interval (1960; Zbl 0093.265)] and \textit{P. G. Nevai} [Orthogonal Polynomials, Mem. Am. Math. Soc. 213 (1979; Zbl 0405.33009)]. These three references, however, show the predicament the specialist in orthogonal polynomials finds himself in nowadays: the interest in the subject has caused an enormous expansion - both in scope and in depth - that it is hardly a matter of one author (and much less of one volume) to cover the entire field, including the most recent results. If we now turn to the subject of asymptotics for orthogonal polynomials and make the restriction to those cases where the support of the measure lies on the real axis only, we still have some ground to cover (just think of what has been done with so-called Freud- weights over the last few years). Nevertheless there is a book that covers it all: the lecture note under review. Leaving out just enough of the (sordid) details, the author has succeeded in writing a book that tells the ''orthogonality adept'' everything he alsways wanted to know. After taking a preliminary course in measure theory and potential theory, it is this monograph that finally gives a textbook-like treatment of the asymptotics for orthogonal polynomials. It might be a idea to consider it for use in graduate course in the future. Let us now cover the contents of the book. After a short introduction (including historical remarks) the first chapter treats the case of a compact support and gives theorems on the behaviour of the zeros (can be used to construct quadrature formulas with simple weights) and n-th root asymptotics (in connection with (logarithmic) capacity). Not only the Szegö-class is treated but also recent results ''beyong the Szegö- class'' are given (Maté, Nevai, Totik etc.). Finally - we leave the real axis here - results are given for orthogonal polynomials on Julia-sets (cf. also Harrington, Bessis, Moussa, Barnsley etc.). The relation with the Frostman-measure plays an important role here. The second chapter looks at the well known three-term recurrence relation for orthogonal polynomials and starts with the case of periodic coefficients (i.e. periodic as functions of their index) and then covers asymptotic periodicity. Theorems are given on the connection between the speed of convergence to the periodic limit-sequence and the existence of analytic solutions of the recurrence relation (strong asymptotics) and in the case of no conditions on the speed of convergence (weak asymptotics). The third chapter is placed in the setting of probability theory. Application of the local limit theorem to the Bernoulli-, Poisson- and negative-binomial distribution makes it possible to give short proofs for known first-order asymptotic results on Jacobi polynomials \(P_ n^{(\alpha,\beta)}\) and Laguerre polynomials \(L_ n^{\alpha}(bx)\) (\(\alpha\) non-negative integer; \(b=n\), \(n+1/2\) or \(n+1;x<0)\). The latter also imply results for Hermite polynomials \(H_ n^{(\alpha)}(iy\sqrt{2n})(\alpha -1/2\) a positive integer; \(y>0)\). However, the probabilistic approach makes it possible to derive asymptotic results that were not previously known (joint work of the author with M. Maejima): Charlier polynomials \(C_ n^{(a)}(bx)\) with \(a>0\), \(x<0\), \(b=n\) or \(n+1\), Meixner polynomials \(m_ n(bx;\beta,c)\) with \(\beta\) a positive integer, \(x<0\), \(b=n\) or \(n+1\). The method can be improved to give asymptotic expansions in the cases of \(P_ n^{(\alpha,\alpha)}(x)\), \(L_ n^{(0)}(nx)\) and \(m_ n(nx;1,c).\) Orthogonal polynomials on unbounded sets are treated in chapter four: the role of the Frostman measure - specifically the arcsin measure on [-1,1] - is now played by the family of Ullman measures on Borel sets in [-1,1]. This chapter contains very advanced material (a.o. connected with Freud's conjecture on weight functions of the form \(| x|^{\beta}\exp (- | x|^{\alpha})\), \(\alpha >0\), \(\beta >-1\) and their generalisations) including results derived by means of the recurrence relation and on the Plancherel-Rotach asymptotics. Finally results on weighted zero distributions are given (most of them due to the author). The fifth chapter gives some consequences of the existence of the asymptotic zero distributions derived in the previous chapters and also covers weak asymptotics for Christoffel functions and results on the associated Toeplitz matrices. Finally, in chapter six, some applications are given, concentrating on less well known ones: the distribution of eigenvalues of certain random \(n\times n\) matrices, discrete Schrödinger equations in one dimension and birth-death processes. In an appendix (6 pages) some background on the Stieltjes-transform and convolution are given, followed by a list of more than 200 references. The author also supplies a 'subject index' (for those impatient readers that want to look up their 'favorite' rightaway, without following the flow of the exposé). The conclusion is obvious: an extremely well-written book, indispensible for people both within and wanting to enter the field of orthogonal polynomials.
scientific article; zbMATH DE number 4000674