On Salikhov’s integral

Sorokin, V.

doi:10.1090/mosc/254

On Salikhov’s integral
HTML articles powered by AMS MathViewer

by V. N. Sorokin
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2016, 107-126

DOI: https://doi.org/10.1090/mosc/254

Published electronically: November 28, 2016

PDF | Request permission

Abstract:

We state a new interpolation problem, which we solve using Salikhov’s integral. This was previously used in the theory of Diophantine approximations. We study the asymptotic behaviour of orthogonal polynomials related to this problem.

References

E. M. Nikishin and V. N. Sorokin, Ratsional′nye approksimatsii i ortogonal′nost′, “Nauka”, Moscow, 1988 (Russian). MR 953788
Kurt Mahler, Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I, J. Reine Angew. Math. 166 (1932), 118–136 (German). MR 1581302, DOI 10.1515/crll.1932.166.118
N. I. Fel′dman, An estimate of the absolute value of a linear form in the logarithms of certain algebraic numbers, Mat. Zametki 2 (1967), 245–256 (Russian). MR 219478
E. M. Nikišin, Correction: “Logarithms of natural numbers”, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 972 (Russian). MR 587346
V. N. Sorokin, Estimates for polynomials in logarithms of some rational numbers, Fundam. Prikl. Mat. 11 (2005), no. 6, 179–194 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 146 (2007), no. 2, 5759–5770. MR 2204431, DOI 10.1007/s10958-007-0390-0
Masayoshi Hata, Rational approximations to $\pi$ and some other numbers, Acta Arith. 63 (1993), no. 4, 335–349. MR 1218461, DOI 10.4064/aa-63-4-335-349
V. N. Sorokin and E. N. Cherednikova, Some remarks on the approximation of logarithms, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (2007), 21–26, 79 (Russian, with Russian summary); English transl., Moscow Univ. Math. Bull. 62 (2007), no. 2, 61–66. MR 2357042, DOI 10.3103/S0027132207020040
E. A. Rukhadze, A lower bound for the approximation of $\textrm {ln}\,2$ by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1987), 25–29, 97 (Russian). MR 922879
V. V. Zudilin, An essay on the irrationality measures of $\pi$ and other logarithms, Chebyshevskiĭ Sb. 5 (2004), no. 2(10), 49–65 (Russian). MR 2140069
V. Kh. Salikhov, On the irrationality measure of $\ln 3$, Dokl. Akad. Nauk 417 (2007), no. 6, 753–755 (Russian); English transl., Dokl. Math. 76 (2007), no. 3, 955–957. MR 2462856, DOI 10.1134/S1064562407060361
N. S. Landkoff, Foundations of modern potential theory, Nauka, Moscow, 1966. (Russian)
A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin, On Hermite-Padé approximants for systems of functions of Markov type, Mat. Sb. 188 (1997), no. 5, 33–58 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 5, 671–696. MR 1478629, DOI 10.1070/SM1997v188n05ABEH000225