Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 004, 48 pages      arXiv:2307.09277      https://doi.org/10.3842/SIGMA.2024.004
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Percy Deift ^a and Mateusz Piorkowski ^b
a) Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Str., New York, NY 10012, USA
^b) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

Received July 19, 2023, in final form January 01, 2024; Published online January 10, 2024

Abstract
We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by Magnus and extends earlier results by Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at $x = +1$.

Key words: orthogonal polynomials; Riemann-Hilbert problems; recurrence coefficients; steepest descent method.

pdf (750 kb)   tex (56 kb)  

References

1. Baratchart L., Yattselev M., Convergent interpolation to Cauchy integrals over analytic arcs with Jacobi-type weights, Int. Math. Res. Not. 2010 (2010), 4211-4275, arXiv:0911.3850.
2. Bleher P., Its A., Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. 150 (1999), 185-266, arXiv:math-ph/9907025.
3. Böttcher A., Karlovich Yu.I., Carleson curves, Muckenhoupt weights, and Toeplitz operators, Progr. Math., Vol. 154, Birkhäuser, Basel, 1997.
4. Conway T.O., Deift P., Asymptotics of polynomials orthogonal with respect to a logarithmic weight, SIGMA 14 (2018), 056, 66 pages, arXiv:1711.01590.
5. Deift P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lect. Notes Math., Vol. 3, American Mathematical Society, Providence, RI, 1999.
6. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335-1425.
7. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491-1552.
8. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295-368.
9. Deift P., Zhou X., Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Comm. Pure Appl. Math. 56 (2003), 1029-1077, arXiv:math.AP/0206222.
10. Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in $2$D quantum gravity, Comm. Math. Phys. 147 (1992), 395-430.
11. Kriecherbauer T., McLaughlin K.T.-R., Strong asymptotics of polynomials orthogonal with respect to Freud weights, Int. Math. Res. Not. 1999 (1999), 299-333.
12. Kuijlaars A.B.J., McLaughlin K.T.-R., Van Assche W., Vanlessen M., The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$, Adv. Math. 188 (2004), 337-398, arXiv:math.CA/0111252.
13. Lenard A., Some remarks on large Toeplitz determinants, Pacific J. Math. 42 (1972), 137-145.
14. Lenells J., Matrix Riemann-Hilbert problems with jumps across Carleson contours, Monatsh. Math. 186 (2018), 111-152, arXiv:1401.2506.
15. Magnus A.P., Gaussian integration formulas for logarithmic weights and application to 2-dimensional solid-state lattices, J. Approx. Theory 228 (2018), 21-57.
16. McLaughlin K.T.-R., Miller P.D., The $\overline{\partial}$ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, Int. Math. Res. Pap. 2006 (2006), 48673, 77 pages, arXiv:math.CA/0406484.
17. McLaughlin K.T.-R., Miller P.D., The $\overline{\partial}$ steepest descent method for orthogonal polynomials on the real line with varying weights, Int. Math. Res. Not. 2008 (2008), rnn075, 66 pages, arXiv:0805.1980.
18. Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A., NIST digital library of mathematical functions, Release 1.1.5 of 2022-03-15, available at http://dlmf.nist.gov/.
19. Piorkowski M., Riemann-Hilbert theory without local parametrix problems: applications to orthogonal polynomials, J. Math. Anal. Appl. 504 (2021), 125495, 23 pages, arXiv:2021.12549.
20. Van Assche W., Hermite-Padé rational approximation to irrational numbers, Comput. Methods Funct. Theory 10 (2010), 585-602.
21. Yattselev M.L., On smooth perturbations of Chebyshëv polynomials and $ \bar {\partial } $-Riemann-Hilbert method, Canad. Math. Bull. 66 (2023), 142-155, arXiv:2202.10374.