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Colloquium of Steklov Mathematical Institute of Russian Academy of Sciences
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Steklov problem and estimates on continuous spectrum A. I. Aptekarev Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow |
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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 $$ M_{n,\delta}=\sup_{\sigma\in S_\delta}\|\phi_n\|_{L^\infty(\mathbb{T})}, $$ where 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 References
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