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On a problem by Steklov

Authors: A. Aptekarev, S. Denisov and D. Tulyakov
Journal: J. Amer. Math. Soc. 29 (2016), 1117-1165
MSC (2010): Primary 42C05; Secondary 33D45
Published electronically: December 24, 2015
MathSciNet review: 3522611
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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 )\ge \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 $ \Vert\phi _n\Vert _{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 }=\smash {\sup \limits _{\sigma \in S_\delta }}\Vert\phi _n\Vert _{L^\infty (\mathbb{T})}$. Then, we prove

$\displaystyle C(\delta )\sqrt n < M_{n,\delta }\le \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.

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  • [1] 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 (92k:42032)
  • [2] 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 (95k:42037),
  • [3] 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 (2010m:81060),
  • [4] 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 (2000g:33001)
  • [5] 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 (2005f:42052),
  • [6] 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.
  • [7] 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.
  • [8] 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,
  • [9] P. Duren, Theory of $ H^p$ Spaces, Dover Publications, Mineola, NY, 2000.
  • [10] 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).
  • [11] Ja. L. Geronimus, Some estimates of orthogonal polynomials and the problem of Steklov, Dokl. Akad. Nauk 236 (1977), no. 1, 14-17 (Russian). MR 0467147 (57 #7013)
  • [12] 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 (28 #3283)
  • [13] Ja. L. Geronimus, On a conjecture of V. A. Steklov, Dokl. Akad. Nauk 142 (1962), 507-509 (Russian). MR 0132965 (24 #A2801)
  • [14] B. L. Golinskiĭ, The problem of V. A. Steklov in the theory of orthogonal polynomials, Mat. Zametki 15 (1974), 21-32 (Russian). MR 0342944 (49 #7688)
  • [15] 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 (2007f:33001)
  • [16] 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,
  • [17] Doron S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Ann. of Math. (2) 170 (2009), no. 2, 915-939. MR 2552113 (2011a:42042),
  • [18] 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 (92i:42014),
  • [19] Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926 (80k:42025),
  • [20] 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 (92k:42034),
  • [21] George Pólya and Gábor Szegő, Problems and theorems in analysis. I, 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; Corrected printing of the revised translation of the fourth German edition. MR 580154 (81e:00002)
  • [22] E. A. Rahmanov (E. A. Rakhmanov), Steklov's conjecture in the theory of orthogonal polynomials, Mat. Sb. (N.S.) 108(150) (1979), no. 4, 581-608, 640 (Russian). MR 534610 (81j:42042)
  • [23] E. A. Rahmanov (E. A. Rakhmanov), 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 (82h:42029b)
  • [24] G. Szegő, Orthogonal Polynomials, (fourth edition), Amer. Math. Soc. Colloq. Publ., vol. 23, American Mathematical Society, Providence, RI, 1975.
  • [25] B. Simon, Orthogonal polynomials on the unit circle, Vols. 1 and 2, American Mathematical Society, Providence, RI, 2005.
  • [26] 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.
  • [27] 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 (58 #12173)
  • [28] Vilmos Totik, Christoffel functions on curves and domains, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2053-2087. MR 2574887 (2011b:30006),
  • [29] Vilmos Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283-303. MR 1785285 (2001j:42021),
  • [30] 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 (2004h:01041)

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Additional Information

A. Aptekarev
Affiliation: Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia

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

D. Tulyakov
Affiliation: Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia

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.
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