Orthogonal polynomials for exponential weights.

*(English)*Zbl 0997.42011
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 4. New York, NY: Springer. xi, 476 p. (2001).

Orthogonal polynomials \(p_n,\;n=0,1,2,\ldots\), have a special status within approximation theory. They are of course relevant for least squares approximation by polynomials and are the building blocks for Fourier analysis with polynomials. They also appear in rational approximation as denominators (and numerators) of Padé approximants [E. M. Nikishin and V. N. Sorokin, “Rational approximations and orthogonality” (1991; Zbl 0733.41001)]. Even for multipoint Padé approximation one gets orthogonal polynomials, but now with a varying weight. Recently, orthogonal rational functions have been investigated [A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad, “Orthogonal rational functions” (1999; Zbl 0923.42017)], and again they give rise to orthogonal polynomials with varying weights. Needless to say that certain continued fractions also give rise to orthogonal polynomials. Orthogonal polynomials very often can be studied as special functions, and in fact most explicit examples appear in the Askey table. This aspect can for instance be found in the book of G. E. Andrews, R. Askey, and R. Roy [“Special functions” (1999; Zbl 0920.33001)]. The book under review, however, is not on special functions but on approximation theory, with emphasis on bounds for weighted orthogonal polynomials. Exponential weights in this context are weights of the form \(w(x) = W^2(x) = \exp(-2 Q(x))\), where \(Q\) has some growth conditions as \(x\) approaches the endpoints of the support, which is the real line or an interval \(I\) of the real line that contains \(0\). The bounds which are investigated in this book are of the form
\[
\sup_{x \in I} |p_n(x)|W(x) |(a_n-x)(x-a_{-n})|^{1/4} \leq C,
\]
where the numbers \(a_n\) and \(a_{-n}\) are very important in the theory developed, and are known as Mhaskar-Rakhmanov-Saff numbers.

This book is not a textbook for use in a (graduate) class in approximation theory. It much more looks like a set of research papers, with very recent results and detailed proofs, using the latest available tools and techniques, such as weighted (logarithmic) potential theory [E. B. Saff and V. Totik, “Logarithmic potentials with external fields” (1997; Zbl 0881.31001)]. Various results on orthogonal polynomials with exponential weights have been published the past 20 years, in particular for Freud weights (where \(Q\) is even and of polynomial growth near \(\pm \infty\)) and for Erdős weights (where \(Q\) is even and of faster than polynomial growth), but also on a finite interval one found new results which did not fit into the theory worked out by Szegő in the first half of the twentieth century. Particularly Pollaczek weights on \([-1,1]\), where \(Q\) grows like \(1/\sqrt{1-x^2}\) near \(\pm 1\), were considered earlier by A. L. Levin and D. S. Lubinsky [“Christoffel functions and orthogonal polynomials for exponential weights on \([-1,1]\)” (1994; Zbl 0810.42012)]. The main achievement of the present book is that it provides a unified treatment of all these cases: finite and infinite intervals and \(Q\) of whatever rate of growth, even without the restriction of evenness. To this end, the authors introduce six classes of weights, which they call \({\mathcal F}(C^2) \subset {\mathcal F}(\operatorname {lip} {1 \over 2}) \subset {\mathcal F}(\operatorname {Lip} {1 \over 2}) \subset {\mathcal F}(\text{dini}) \subset {\mathcal F}(\text{Dini}) \subset {\mathcal F}\). These classes are a bit technical, so it is good to keep in mind some specific examples, such as the Freud weights \(Q(x) = |x|^\alpha\), with \(\alpha > 1\), which are in \({\mathcal F}(C^2)\), and the Erdős weight \(Q(x)=\exp(x)-1\), or even higher order iterates of the exponential function, which are also in \({\mathcal F}(C^2)\), and for a finite interval, a typical example is \(Q(x) = (1-x^2)^{-\alpha}-1\) on \([-1,1]\).

In Chapter 1 the classes of weights are introduced and the main results in the book are already formulated. The basic properties of the Mhaskar-Rakhmanov-Saff numbers and of the function \(Q\) are worked out in Chapters 2 and 3. Restricted range inequalities, or infinite-finite range inequalities, are proved in Chapter 4, not only for (weighted) polynomials but also for exponentials of potentials. Such inequalities show that weighted polynomials on \(I\) live on a smaller interval \([a_{-n},a_n]\), where the endpoints are the Mhaskar-Rakhmanov-Saff numbers. These proofs involve potential theory, and in particular an important density function \(\sigma_t(x)\) defined in terms of the quantities \(a_{\pm t}\) and the function \(Q\). Estimates for this density and its potential are given in Chapter 5 and smoothness properties in Chapter 6. Chapter 7 establishes the existence of weighted polynomial approximations, which are needed for the estimates and the asymptotics in the remaining chapters. Estimates of the Christoffel function are proved in Chapter 9 and Markov-Bernstein inequalities are given in Chapter 10. Zeros of orthogonal polynomials are investigated in Chapter 11, in particular the largest zeros and the spacing of the zeros. The orthogonal polynomials themselves are studied in Chapter 12, which mainly deals with bounds on the (weighted) orthogonal polynomials. Asymptotic results can be found in Chapter 8, which deals with extremal errors, and Chapters 14 (extremal polynomials) and 15 (orthonormal polynomials).

This book is a must for approximators, in particularly those interested in weighted polynomial approximation or orthogonal polynomials. It cannot serve as a textbook but will probably be indispensible for research in this field, since all the important tools, results, properties are there, with detailed proofs and to the point references.

This book is not a textbook for use in a (graduate) class in approximation theory. It much more looks like a set of research papers, with very recent results and detailed proofs, using the latest available tools and techniques, such as weighted (logarithmic) potential theory [E. B. Saff and V. Totik, “Logarithmic potentials with external fields” (1997; Zbl 0881.31001)]. Various results on orthogonal polynomials with exponential weights have been published the past 20 years, in particular for Freud weights (where \(Q\) is even and of polynomial growth near \(\pm \infty\)) and for Erdős weights (where \(Q\) is even and of faster than polynomial growth), but also on a finite interval one found new results which did not fit into the theory worked out by Szegő in the first half of the twentieth century. Particularly Pollaczek weights on \([-1,1]\), where \(Q\) grows like \(1/\sqrt{1-x^2}\) near \(\pm 1\), were considered earlier by A. L. Levin and D. S. Lubinsky [“Christoffel functions and orthogonal polynomials for exponential weights on \([-1,1]\)” (1994; Zbl 0810.42012)]. The main achievement of the present book is that it provides a unified treatment of all these cases: finite and infinite intervals and \(Q\) of whatever rate of growth, even without the restriction of evenness. To this end, the authors introduce six classes of weights, which they call \({\mathcal F}(C^2) \subset {\mathcal F}(\operatorname {lip} {1 \over 2}) \subset {\mathcal F}(\operatorname {Lip} {1 \over 2}) \subset {\mathcal F}(\text{dini}) \subset {\mathcal F}(\text{Dini}) \subset {\mathcal F}\). These classes are a bit technical, so it is good to keep in mind some specific examples, such as the Freud weights \(Q(x) = |x|^\alpha\), with \(\alpha > 1\), which are in \({\mathcal F}(C^2)\), and the Erdős weight \(Q(x)=\exp(x)-1\), or even higher order iterates of the exponential function, which are also in \({\mathcal F}(C^2)\), and for a finite interval, a typical example is \(Q(x) = (1-x^2)^{-\alpha}-1\) on \([-1,1]\).

In Chapter 1 the classes of weights are introduced and the main results in the book are already formulated. The basic properties of the Mhaskar-Rakhmanov-Saff numbers and of the function \(Q\) are worked out in Chapters 2 and 3. Restricted range inequalities, or infinite-finite range inequalities, are proved in Chapter 4, not only for (weighted) polynomials but also for exponentials of potentials. Such inequalities show that weighted polynomials on \(I\) live on a smaller interval \([a_{-n},a_n]\), where the endpoints are the Mhaskar-Rakhmanov-Saff numbers. These proofs involve potential theory, and in particular an important density function \(\sigma_t(x)\) defined in terms of the quantities \(a_{\pm t}\) and the function \(Q\). Estimates for this density and its potential are given in Chapter 5 and smoothness properties in Chapter 6. Chapter 7 establishes the existence of weighted polynomial approximations, which are needed for the estimates and the asymptotics in the remaining chapters. Estimates of the Christoffel function are proved in Chapter 9 and Markov-Bernstein inequalities are given in Chapter 10. Zeros of orthogonal polynomials are investigated in Chapter 11, in particular the largest zeros and the spacing of the zeros. The orthogonal polynomials themselves are studied in Chapter 12, which mainly deals with bounds on the (weighted) orthogonal polynomials. Asymptotic results can be found in Chapter 8, which deals with extremal errors, and Chapters 14 (extremal polynomials) and 15 (orthonormal polynomials).

This book is a must for approximators, in particularly those interested in weighted polynomial approximation or orthogonal polynomials. It cannot serve as a textbook but will probably be indispensible for research in this field, since all the important tools, results, properties are there, with detailed proofs and to the point references.

Reviewer: Walter Van Assche (Leuven)

##### MSC:

42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |

41A10 | Approximation by polynomials |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

41-02 | Research exposition (monographs, survey articles) pertaining to approximations and expansions |