General orthogonal polynomials.

*(English)*Zbl 0791.33009
Encyclopedia of Mathematics and Its Applications. 43. Cambridge: Cambridge University Press. xii, 250 p. (1992).

[The following review has been adapted from the preface of the book.]

The theory of orthogonal polynomials can be divided into two loosely related parts. One of them is the formal, algebraic aspect of the theory, which has close connections with special functions, combinatorics, and algebra. The investigation of more general orthogonal polynomials with methods of mathematical analysis belongs to the other part of the theory. Here the central questions are the asymptotic behavior of the polynomials and their zeros, recovering the measure of orthogonality, and so forth.

This book is exclusively devoted to the second part of the theory. The main emphasis is on the investigation of the asymptotic behavior of general orthogonal polynomials, but related questions as, for instance, the distribution of zeros are also taken into consideration.

Until recently most of the asymptotic theory of orthogonal polynomials has concentrated on orthogonal systems for which the measure of orthogonality is supported on the real or on the unit circle. Even then it has been usually assumed that the measure of orthogonality is sufficiently thick on its support. This book is devoted to orthogonal polynomials with respect to general measures \(\mu\). The only requirement on \(\mu\) is that it has compact support on \(\mathbb{C}\), that is, both the support \(S(\mu)\) of \(\mu\) and the “thickness” of \(\mu\) can be arbitrary. Thus both the measure and its support can be “wild,” each of which has its own reflection in the general theory.

For orthogonal polynomials \(p_ n(\mu;z)\), as for sequences of polynomials in general, there exists a hierarchy of types of asymptotic behavior. We mention here the most common ones, which are called power (or Szegö), ratio, and \(n\)-th-root asymptotic behavior. Roughly speaking, these mean that the sequences \[ \text{(P.1)} \left\{{p_ n(\mu;z)\over f(z)^ n}:n\in \mathbb{N}\right\},\quad\text{(P.2)} \left\{{p_{n+1}(\mu/z)\over p_ n(\mu;z)}:n\in \mathbb{N}\right\},\quad\text{(P.3)} \left\{\root n\of{p_ n(\mu;z)}:n\in \mathbb{N}\right\}, \] respectively, tend to a limit on a certain set of values \(z\in \mathbb{C}\) as \(n\to\infty\). Since each type of asymptotics in the hierarchy (P.1) to (P.3) implies the next one, the \(n\)-th-root asymptotic behavior is the most general of the three types and requires the weakest assumptions. At the same time it is sufficient for many applications, as, for instance, the convergence of polynomial (Chebyshev-Fourier) expansions based on the system \(\{p_ n(\mu;z):n\in \mathbb{N}\}\), or the convergence of continued fractions or Padé approximants to Markov functions.

In this context the present work can be classified as a monograph on the \(n\)-th-root asymptotic behavior. Earlier research in this direction has been mainly due to P. P. Korovkin, J. Ullman, P. Erdős, G. Freud, P. Turán and H. Widom, although the case of general support has barely been touched upon. It was especially J. Ullman who systematically studied different bounds and asymptotics on orthogonal polynomials with respect to arbitrary measures \(\mu\) on \([-1,1]\). This monograph synthesizes and considerably extends earlier research concerning general orthogonal polynomials. A large part of it contains new results very often without any precedence. Special emphasis has been placed on examples illustrating that the results are sharp. The last chapter is devoted to applications.

The proofs use potential-theoretical considerations. The usefulness of logarithmic potentials in the general theory can be easily understood if one recalls that the modulus of a polynomial is basically nothing else than the exponential of a discrete potential. Most proofs are based on \(L^ 2\) minimality of the monic orthogonal polynomials, not on the orthogonality property, hence the method works for \(L^ p\)-extremal polynomials as well.

The content of the different chapters is briefly as follows. Chapter 1 gives sharp upper and lower bounds for orthogonal polynomials and their leading coefficients. Chapter 2 examines the location and asymptotic distribution of the zeros. An extremely important concept, “regular \((n\)-th-root) asymptotic behavior” (in symbols, \(\mu\in\mathbf{Reg})\), is introduced and characterized in chapter 3. Polynomials with this property are the natural analogue of classical orthogonal polynomials in the general case, and they have many applications and equivalent formulations in different subjects of approximation theory. To facilitate these applications one needs easy-to-use criteria for \(\mu\in\mathbf{Reg}\), which are given in chapter 4. In chapter 5 a surprising phenomenon is investigated: regularity is basically a local property. Finally, chapter 6 contains several applications of \(\mu\in\mathbf{Reg}\), and we list such two examples: (1) The classical connection between continued fractions and orthogonal polynomials is extended to rational interpolation and best rational approximation of Cauchy transforms of measures \(\mu\). (2) It is shown that \(\mu\in\mathbf{Reg}\) is equivalent to an exact maximal rate of convergence for these rational interpolants or approximants. An appendix includes those results from the theory of logarithmic potentials that are frequently used in the text.

The theory of orthogonal polynomials can be divided into two loosely related parts. One of them is the formal, algebraic aspect of the theory, which has close connections with special functions, combinatorics, and algebra. The investigation of more general orthogonal polynomials with methods of mathematical analysis belongs to the other part of the theory. Here the central questions are the asymptotic behavior of the polynomials and their zeros, recovering the measure of orthogonality, and so forth.

This book is exclusively devoted to the second part of the theory. The main emphasis is on the investigation of the asymptotic behavior of general orthogonal polynomials, but related questions as, for instance, the distribution of zeros are also taken into consideration.

Until recently most of the asymptotic theory of orthogonal polynomials has concentrated on orthogonal systems for which the measure of orthogonality is supported on the real or on the unit circle. Even then it has been usually assumed that the measure of orthogonality is sufficiently thick on its support. This book is devoted to orthogonal polynomials with respect to general measures \(\mu\). The only requirement on \(\mu\) is that it has compact support on \(\mathbb{C}\), that is, both the support \(S(\mu)\) of \(\mu\) and the “thickness” of \(\mu\) can be arbitrary. Thus both the measure and its support can be “wild,” each of which has its own reflection in the general theory.

For orthogonal polynomials \(p_ n(\mu;z)\), as for sequences of polynomials in general, there exists a hierarchy of types of asymptotic behavior. We mention here the most common ones, which are called power (or Szegö), ratio, and \(n\)-th-root asymptotic behavior. Roughly speaking, these mean that the sequences \[ \text{(P.1)} \left\{{p_ n(\mu;z)\over f(z)^ n}:n\in \mathbb{N}\right\},\quad\text{(P.2)} \left\{{p_{n+1}(\mu/z)\over p_ n(\mu;z)}:n\in \mathbb{N}\right\},\quad\text{(P.3)} \left\{\root n\of{p_ n(\mu;z)}:n\in \mathbb{N}\right\}, \] respectively, tend to a limit on a certain set of values \(z\in \mathbb{C}\) as \(n\to\infty\). Since each type of asymptotics in the hierarchy (P.1) to (P.3) implies the next one, the \(n\)-th-root asymptotic behavior is the most general of the three types and requires the weakest assumptions. At the same time it is sufficient for many applications, as, for instance, the convergence of polynomial (Chebyshev-Fourier) expansions based on the system \(\{p_ n(\mu;z):n\in \mathbb{N}\}\), or the convergence of continued fractions or Padé approximants to Markov functions.

In this context the present work can be classified as a monograph on the \(n\)-th-root asymptotic behavior. Earlier research in this direction has been mainly due to P. P. Korovkin, J. Ullman, P. Erdős, G. Freud, P. Turán and H. Widom, although the case of general support has barely been touched upon. It was especially J. Ullman who systematically studied different bounds and asymptotics on orthogonal polynomials with respect to arbitrary measures \(\mu\) on \([-1,1]\). This monograph synthesizes and considerably extends earlier research concerning general orthogonal polynomials. A large part of it contains new results very often without any precedence. Special emphasis has been placed on examples illustrating that the results are sharp. The last chapter is devoted to applications.

The proofs use potential-theoretical considerations. The usefulness of logarithmic potentials in the general theory can be easily understood if one recalls that the modulus of a polynomial is basically nothing else than the exponential of a discrete potential. Most proofs are based on \(L^ 2\) minimality of the monic orthogonal polynomials, not on the orthogonality property, hence the method works for \(L^ p\)-extremal polynomials as well.

The content of the different chapters is briefly as follows. Chapter 1 gives sharp upper and lower bounds for orthogonal polynomials and their leading coefficients. Chapter 2 examines the location and asymptotic distribution of the zeros. An extremely important concept, “regular \((n\)-th-root) asymptotic behavior” (in symbols, \(\mu\in\mathbf{Reg})\), is introduced and characterized in chapter 3. Polynomials with this property are the natural analogue of classical orthogonal polynomials in the general case, and they have many applications and equivalent formulations in different subjects of approximation theory. To facilitate these applications one needs easy-to-use criteria for \(\mu\in\mathbf{Reg}\), which are given in chapter 4. In chapter 5 a surprising phenomenon is investigated: regularity is basically a local property. Finally, chapter 6 contains several applications of \(\mu\in\mathbf{Reg}\), and we list such two examples: (1) The classical connection between continued fractions and orthogonal polynomials is extended to rational interpolation and best rational approximation of Cauchy transforms of measures \(\mu\). (2) It is shown that \(\mu\in\mathbf{Reg}\) is equivalent to an exact maximal rate of convergence for these rational interpolants or approximants. An appendix includes those results from the theory of logarithmic potentials that are frequently used in the text.

Reviewer: M.Wyneken (Flint)