## On a problem by Steklov

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- by A. Aptekarev, S. Denisov and D. Tulyakov PDF
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## Abstract:

Given any $\delta \in (0,1)$, we define the Steklov class $S_\delta$ to be the set of probability measures $\sigma$ on the unit circle $\mathbb {T}$, such that $\sigma ’(\theta )\geqslant \delta /(2\pi )>0$ for Lebesgue almost every $\theta \in [0,2\pi )$. One can define the orthonormal polynomials $\phi _n(z)$ with respect to $\sigma \in S_\delta$. In this paper, we obtain the sharp estimates on the uniform norms $\|\phi _n\|_{L^\infty (\mathbb T)}$ as $n\to \infty$ which settles a question asked by Steklov in 1921. As an important intermediate step, we consider the following variational problem. Fix $n\in \mathbb N$ and define $M_{n,\delta }=\sup \limits _{\sigma \in S_\delta }\|\phi _n\|_{L^\infty (\mathbb T)}$. Then, we prove \[ C(\delta )\sqrt n < M_{n,\delta }\leqslant \sqrt {\frac {n+1}\delta } . \] A new method is developed that can be used to study other important variational problems. For instance, we prove the sharp estimates for the polynomial entropy in the Steklov class.## References

- M. U. Ambroladze,
*On the possible rate of growth of polynomials that are orthogonal with a continuous positive weight*, Mat. Sb.**182**(1991), no. 3, 332–353 (Russian); English transl., Math. USSR-Sb.**72**(1992), no. 2, 311–331. MR**1110069**, DOI 10.1070/SM1992v072n02ABEH001269 - A. I. Aptekarev, V. S. Buyarov, and I. S. Degeza,
*Asymptotic behavior of $L^p$-norms and entropy for general orthogonal polynomials*, Mat. Sb.**185**(1994), no. 8, 3–30 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math.**82**(1995), no. 2, 373–395. MR**1302621**, DOI 10.1070/SM1995v082n02ABEH003571 - A. I. Aptekarev, J. S. Dehesa, and A. Martinez-Finkelshtein,
*Asymptotics of orthogonal polynomial’s entropy*, J. Comput. Appl. Math.**233**(2010), no. 6, 1355–1365. MR**2559324**, DOI 10.1016/j.cam.2009.02.056 - George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958**, DOI 10.1017/CBO9781107325937 - B. Beckermann, A. Martínez-Finkelshtein, E. A. Rakhmanov, and F. Wielonsky,
*Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class*, J. Math. Phys.**45**(2004), no. 11, 4239–4254. MR**2098132**, DOI 10.1063/1.1794842 - S. Bernstein,
*Sur les polynomes orthogonaux relatifs $\grave {\rm {a}}$ un segment fini*, J. Mathem$\acute {\rm {a}}$tiques,**9**(1930), no. 4, 127–177. 10 (1931), pp. 219–286. - S. Denisov,
*On the size of the polynomials orthonormal on the unit circle with respect to a measure which is a sum of the Lebesgue measure and $p$ point masses*. To appear in Proceedings of the AMS. - S. Denisov and S. Kupin,
*On the growth of the polynomial entropy integrals for measures in the Szegő class*, Adv. Math.**241**(2013), 18–32. MR**3053702**, DOI 10.1016/j.aim.2013.03.014 - P. Duren,
*Theory of $H^p$ Spaces*, Dover Publications, Mineola, NY, 2000. - Ya. L. Geronimus,
*Polynomials Orthogonal on the Circle and on the Interval*, International Series of Monographs on Pure and Applied Mathematics, vol. 18, Pergamon Press, New York-Oxford-London-Paris, 1960. GIFML, Moscow, 1958 (in Russian). - Ja. L. Geronīmus,
*Some estimates of orthogonal polynomials and the problem of Steklov*, Dokl. Akad. Nauk SSSR**236**(1977), no. 1, 14–17 (Russian). MR**0467147** - Ja. L. Geronimus,
*The relation between the order of growth of orthonormal polynomials and their weight function*, Mat. Sb. (N.S.)**61 (103)**(1963), 65–79 (Russian). MR**0160069** - Ja. L. Geronimus,
*On a conjecture of V. A. Steklov*, Dokl. Akad. Nauk SSSR**142**(1962), 507–509 (Russian). MR**0132965** - B. L. Golinskiĭ,
*The problem of V. A. Steklov in the theory of orthogonal polynomials*, Mat. Zametki**15**(1974), 21–32 (Russian). MR**342944** - Mourad E. H. Ismail,
*Classical and quantum orthogonal polynomials in one variable*, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR**2191786**, DOI 10.1017/CBO9781107325982 - A. Kroó and D. S. Lubinsky,
*Christoffel functions and universality in the bulk for multivariate orthogonal polynomials*, Canad. J. Math.**65**(2013), no. 3, 600–620. MR**3043043**, DOI 10.4153/CJM-2012-016-x - Doron S. Lubinsky,
*A new approach to universality limits involving orthogonal polynomials*, Ann. of Math. (2)**170**(2009), no. 2, 915–939. MR**2552113**, DOI 10.4007/annals.2009.170.915 - Attila Máté, Paul Nevai, and Vilmos Totik,
*Szegő’s extremum problem on the unit circle*, Ann. of Math. (2)**134**(1991), no. 2, 433–453. MR**1127481**, DOI 10.2307/2944352 - Paul G. Nevai,
*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, DOI 10.1090/memo/0213 - Paul Nevai, John Zhang, and Vilmos Totik,
*Orthogonal polynomials: their growth relative to their sums*, J. Approx. Theory**67**(1991), no. 2, 215–234. MR**1133061**, DOI 10.1016/0021-9045(91)90019-7 - George Pólya and Gábor Szegő,
*Problems and theorems in analysis. I*, Corrected printing of the revised translation of the fourth German edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 193, Springer-Verlag, Berlin-New York, 1978. Series, integral calculus, theory of functions; Translated from the German by D. Aeppli. MR**580154** - E. A. Rahmanov,
*Steklov’s conjecture in the theory of orthogonal polynomials*, Mat. Sb. (N.S.)**108(150)**(1979), no. 4, 581–608, 640 (Russian). MR**534610** - E. A. Rahmanov,
*Estimates of the growth of orthogonal polynomials whose weight is bounded away from zero*, Mat. Sb. (N.S.)**114(156)**(1981), no. 2, 269–298, 335 (Russian). MR**609291** - G. Szegő,
*Orthogonal Polynomials*, (fourth edition), Amer. Math. Soc. Colloq. Publ., vol. 23, American Mathematical Society, Providence, RI, 1975. - B. Simon,
*Orthogonal polynomials on the unit circle, Vols. 1 and 2*, American Mathematical Society, Providence, RI, 2005. - V. A. Steklov,
*Une methode de la solution du probleme de development des fonctions en series de polynomes de Tchebysheff independante de la theorie de fermeture*, Izv. Rus. Ac. Sci. (1921), 281–302, 303–326. - P. K. Suetin,
*V. A. Steklov’s problem in the theory of orthogonal polynomials*, Mathematical analysis, Vol. 15 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1977, pp. 5–82 (Russian). MR**0493142** - Vilmos Totik,
*Christoffel functions on curves and domains*, Trans. Amer. Math. Soc.**362**(2010), no. 4, 2053–2087. MR**2574887**, DOI 10.1090/S0002-9947-09-05059-4 - Vilmos Totik,
*Asymptotics for Christoffel functions for general measures on the real line*, J. Anal. Math.**81**(2000), 283–303. MR**1785285**, DOI 10.1007/BF02788993 - A. Zygmund,
*Trigonometric series. Vol. I, II*, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. With a foreword by Robert A. Fefferman. MR**1963498**

## Additional Information

**A. Aptekarev**- Affiliation: Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia
- MR Author ID: 192572
- Email: aptekaa@keldysh.ru
**S. Denisov**- Affiliation: Mathematics Department, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706; and Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia
- MR Author ID: 627554
- Email: denissov@wisc.edu
**D. Tulyakov**- Affiliation: Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia
- MR Author ID: 632175
- Email: dntulyakov@gmail.com
- Received by editor(s): March 17, 2014
- Received by editor(s) in revised form: November 12, 2014, July 26, 2015, and October 12, 2015
- Published electronically: December 24, 2015
- Additional Notes: The work on section 3, which was added in the second revision, July 26, 2015, was supported by Russian Science Foundation grant RSCF-14-21-00025. The research of the first and the third authors on the rest of the paper was supported by Grants RFBR 13-01-12430 OFIm, RFBR 14-01-00604 and Program DMS RAS. The work of the second author on the rest of the paper was supported by NSF Grants DMS-1067413, DMS-1464479.
- © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 1117-1165 - MSC (2010): Primary 42C05; Secondary 33D45
- DOI: https://doi.org/10.1090/jams/853
- MathSciNet review: 3522611