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Colloquium of the Steklov Mathematical Institute of Russian Academy of Sciences
September 29, 2016 17:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

Steklov problem and estimates on continuous spectrum

A. I. Aptekarev

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
 Video records: MP4 2,243.7 Mb MP4 324.3 Mb

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Abstract: A Steklov's problem is to obtain the bounds on the sequence of orthonormal polynomials at the support of the weight of orthogonality. In 1921 [1] V.A. Steklov made a conjecture that if weight of orthogonality is strictly positive then sequence of orthonormal polynomials (at the support of the weight) is bounded.
In 1979 [2] E.A. Rakhmanov disproved this conjecture by constructing the sequence of polynomials orthonormal with respect to a positive weight which has the logarithmical rate of growth. Then the Steklov's problem becomes: to obtain the maximal possible rate of growth for these sequences. The modern version of the Steklov's problem is intimately related with the following extremal problem. For a fixed $n\in\mathbb{N}$, find
$$M_{n,\delta}=\sup_{\sigma\in S_\delta}\|\phi_n\|_{L^\infty(\mathbb{T})},$$
where $\phi_n(z)$ is the orthonormal polynomials with respect to the measure $\sigma \in S_\delta$, and $S_\delta$ is the Steklov's class of probability measures $\sigma$ on the unit circle, such that $\sigma'(\theta)\geqslant\delta/(2\pi)>0$ at every Lebesgue point of $\sigma$.
There is an elementary bound
$$M_n\lesssim\sqrt{n}.$$
In 1981 [3] E.A. Rakhmanov have proved:
$$M_n\gtrsim\sqrt{n}/(\ln n)^{\frac3{2}}.$$
In our joint paper with S.A. Denisov and D.N. Tulyakov [4] we have proved, that
$$M_n\gtrsim\sqrt{n},$$
i.e. the elementary upper bound is sharp.
In the talk we discuss the history, statement of the problem and details of the construction of the extremal orthonormal polynomial. We also consider the Steklov problem in $L_p$, $A_p$ spaces and will touch a connection with the famous nonlinear Carleson conjecture (https://terrytao.wordpress.com/2007/12/17).

References
1. W. Stekloff, “Une méthode de la solution du problème de développement des fonctions en séries de polynomes de Tchébychef indépendante de la théorie de fermeture”, Izvѣstiya Rossiiskoi Akademii Nauk'. VI seriya, 15 (1921), 281–302    ; 303–326
2. E. A. Rakhmanov, “O gipoteze Steklova v teorii ortogonalnykh mnogochlenov”, Matem. sb., 108(150):4 (1979), 581–608      ; E. A. Rakhmanov, “On Steklov's conjecture in the theory of orthogonal polynomials”, Math. USSR-Sb., 36:4 (1980), 549–575
3. E. A. Rakhmanov, “Ob otsenkakh rosta ortogonalnykh mnogochlenov, ves kotorykh otgranichen ot nulya”, Matem. sb., 114(156):2 (1981), 269–298      ; E. A. Rakhmanov, “Estimates of the growth of orthogonal polynomials whose weight is bounded away from zero”, Math. USSR-Sb., 42:2 (1982), 237–263
4. A. Aptekarev, S. Denisov, D. Tulyakov, “On a problem by Steklov”, J. Amer. Math. Soc., 29:4 (2016), 1117–1165, arXiv: 1402.1145

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