The origins of the theory of orthogonal polynomials go back at least to the 18th century when they were studied in terms of continued fractions. The theory is now large and complex: a crossroad of several important domains of analysis such as analytic function theory, analytic theory of differential equations, Fourier and harmonic analysis, spectral theory of Sturm-Liouville operators, and approximation and interpolation, among others.

Autorentext

Evguenii Rakhmanov, University of South Florida, USA.

Inhalt

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