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Vol 213, No 11 (2022)
- Year: 2022
- Articles: 8
- URL: https://journals.rcsi.science/0368-8666/issue/view/7497
Editorial Preface
Matematicheskii Sbornik. 2022;213(11):3-4
3-4
Necessary and sufficient conditions for extending a function to a Caratheodory function
Abstract
A criterion deciding whether a function given by its values (with multiplicities) at a sequence of points in the disc $\mathbb D=\{|z|<1\}$ can be extended to a holomorphic function with nonnegative real part in $\mathbb D$ is stated and proved. In the case when this function is given by the values of its derivatives at $z=0$, this is the well-known Caratheodory criterion. It is also shown that Caratheodory's criterion is a consequence of Schur's criterion and, conversely, Schur's criterion follows from Caratheodory's.Bibliography: 10 titles.
Matematicheskii Sbornik. 2022;213(11):5-24
5-24
25-30
On zeros, bounds, and asymptotics for orthogonal polynomials on the unit circle
Abstract
Let $\mu$ be a measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\{\varphi_{n}\}$ be the orthonormal polynomials for $\mu$ and $ż_{jn}\}$ their zeros. Let $\mu$ be absolutely continuous in an arc $\Delta$ of the unit circle, with $\mu'$ positive and continuous there. We show that uniform boundedness of the orthonormal polynomials in subarcs $\Gamma$ of $\Delta$ is equivalent to certain asymptotic behaviour of their zeros inside sectors that rest on $\Gamma$. Similarly the uniform limit $\lim_{n\to \infty}|\varphi_{n}(z)|^{2}\mu'(z)=1$ is equivalent to related asymptotics for the zeros in such sectors. Bibliography: 27 titles.
Matematicheskii Sbornik. 2022;213(11):31-49
31-49
Counting lattice triangulations: Fredholm equations in combinatorics
Abstract
Let $f(m,n)$ be the number of primitive lattice triangulations of an $m\times n$ rectangle. We compute the limits $\lim_n f(m,n)^{1/n}$ for $m=2,3$. For $m=2$ we obtain the exact value of the limit, which is $(611+\sqrt{73})/36$. For $m=3$ we express the limit in terms of a certain Fredholm integral equation for generating functions. This provides a polynomial-time algorithm (with respect to the number of computed digits) for the computation of the limit with any prescribed precision. Bibliography: 13 titles.
Matematicheskii Sbornik. 2022;213(11):50-78
50-78
A generalization of the discrete Rodrigues formula for Meixner polynomials
Abstract
A generalization of Meixner polynomials leading to a new construction of Apery approximations is put forward. The limiting distribution of the zeros of scaled polynomials is described in terms of algebraic functions. The resulting distribution is shown to be a solution of some vector equilibrium problem in the theory of logarithmic potential. Bibliography: 21 titles.
Matematicheskii Sbornik. 2022;213(11):79-101
79-101
A direct proof of Stahl's theorem for a generic class of algebraic functions
Abstract
Under the assumption that Stahl's $S$-compact set exists we give a short proof of the limiting distribution of the zeros of Pade polynomials and the convergence in capacity of diagonal Pade approximants for a generic class of algebraic functions. The proof is direct, rather than by contradiction as Stahl's original proof was. The ‘generic class’ means, in particular, that all the ramification points of the multisheeted Riemann surface of the algebraic function in question are of the second order (that is, all branch points of the function are of square root type). As a consequence, a conjecture of Gonchar relating to Pade approximations is proved for this class of algebraic functions. We do not use the relations of orthogonality for Pade polynomials. The proof is based on the maximum principle only. Bibliography: 19 titles.
Matematicheskii Sbornik. 2022;213(11):102-117
102-117
On holomorphic mappings of strictly pseudoconvex domains
Abstract
We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with boundaries of class $C^2$. In the second part of the paper we establish an extension of the Wong-Rosay theorem to piecewise smooth strictly pseudoconvex domains. Bibliography: 37 titles.
Matematicheskii Sbornik. 2022;213(11):118-142
118-142