Orthogonal Polynomials
De Gruyter (Verlag)
978-3-11-031385-7 (ISBN)
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The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
Evguenii Rakhmanov, University of South Florida, USA.
Proposed content
1. Elementary introduction
1.1. Orthogonal polynomials on R.
1.2. Pade Approximants.
1.3. Pade Approximants for moment series. Markov theorem.
1.4. Asymptotic moment series. Stieltjes theorem.
1.5. Continued fractions.
1.6. Moment problems.
2. Classical orthogonal polynomials
2.1. Classical weights and classical orthogonal polynomials.
2.2. Darboux formula. Generating functions.
2.3. Differential equations.
2.4. Electrostatic interpretation of zeros of classical OP.
2.5. Lioulille - Green asymptotics.
2.6. Asymptotics for classical orthogonal polynomials.
3. Polynomials orthogonal on the unit circle
3.1. Formal properties. Connection with OP on an interval.
3.2. H2-space in a disc. Boundary values. Szego function.
3.3. Szego asymptotic formula.
3.4. Steklov problem.
3.5. Ratio asymptotis.
3.6. Corollaries for OP on the interval.
3.7. Orthogonal Polynomials on more generals sets in plane.
4. Equilibrium measure and exponential weights on R
4.1. Logarithmic potential and logarithmic energy.
4.2. Equilibrium measure in the external field on R.
4.3. Zero disribution of extremal polynomials.
4.4. Logarithmic asymptotics for OP on real line and the rate of convergence in Stieltjes theorem.
4.5. Discrete approximation of a measure on R.
4.6. Strong asymptotics for Freud-type OP.
5. Discrete orthigonal polynomials
5.1. Constrained equilibrium measure.
5.2. Bounds for polynomials with a unit discrete norm.
5.3. Zero distribution for discrete orthogonal polynomials.
5.4. Some results on strong asymptotics for discrete OP.
6. Polynomials orthogonal on several intervals
6.1. Algebraic Riemann surfaces.
6.2. Green functions and harmonic measures (Abel integrals).
6.3. Abel theorem and Jacobi inversion problem.
6.4. Akhiezer-Widom asymptotic formula. Outline of the proof.
6.5. Extremal problems in multiconneted domains.
6.6. Faber polynomials and discretization of equilibrium measure - two ways to complete the proof af asymptotic formula.
6.7. Hermite-Pade polynomials for Markov-type functions.
6.8. From hyperelliptic to general Riemann surfaces.
6.9. Some results and conjectures on asymptotics.
7. Complex orthogonal polynomials
7.1. Pade approximants for functions with branch points.
7.2. Quadratic differentials.
7.3. Existence theorem for minimal capacity cuts.
7.4. Stahl's theorem.
7.5. Existence of S-curves in harmonic external fields.
7.6. Zero distribution of complex orthogonal polynomials.
7.7. Hermit-Pade approximants for functions with branch points.
7.8. Vector-equilibrium problems and Riemann surfaces.
7.9. Asymptotics for Hermit-Pade polynomials.
8. Random matrices and determinantial processes
| Erscheint lt. Verlag | 1.3.2026 |
|---|---|
| Reihe/Serie | De Gruyter Studies in Mathematics ; 63 |
| Verlagsort | Berlin/Boston |
| Sprache | englisch |
| Maße | 170 x 240 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik | |
| Schlagworte | Analytic Function Theory • Approximation • DGPH • Harmonic Analysis • Interpolation • kuendigung • Orthogonale Polynome • orthogonal polynomials • Personalabbau • REISERER • Sturm-Liouville operators |
| ISBN-10 | 3-11-031385-5 / 3110313855 |
| ISBN-13 | 978-3-11-031385-7 / 9783110313857 |
| Zustand | Neuware |
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