VLADIMIR IGOREVICH PARUSNIKOV
https://doi.org/10.22405/2226-8383-2016-17-1-286-298
Abstract
Vladimir Igorevich Parusnikov died on August 22, 2015 after a long and serious illness. He was born in Moscow on January 21, 1957. Vladimir Igorevich graduated from the Department of Mechanics and Mathematics of the Lomonosov Moscow State University, finished his postgraduate studies in 1982 and he defended his thesis in 1983 in there. At the Keldysh Institute of Applied Mathematics he worked since 1982 as a junior researcher, and since 1996 — as a senior researcher. He has published more than 45 scientific papers on functional and numerical continued fractions and their generalizations.V. I. Parusnikov was kind, honest, responsible man and a talented mathematician. His passing is a great loss for the KIAM and for science in general. He was buried at Khovanskoye cemetery in Moscow. We provide a survey of his mathematical work. First, he studied the generalized functional continued fractions and got rather strong results. The last twenty years he, together with A. D. Bruno, was looking for a multi-dimensional generalization of the continued fraction giving the best Diophantine approximations and the periods in the algebraic case. Such a generalization was finally found. The first Section of the paper was written by A. I. Aptekarev, the second Section was written by A. D. Bruno, the list of scientific publications of V. I. Parusnikov was prepared by A. B. Batkhin.
About the Authors
A. I. Aptekarev
Keldysh Institute of Applied Mathematics
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor,
acting director
A. B. Batkhin
Keldysh Institute of Applied Mathematics
Russian Federation
Russian Federation
Candidate of Physical and Mathematical Sciences,
Associate Professor, Senior Researcher
A. D. Bruno
Keldysh Institute of Applied Mathematics
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor,
acting head of sector
References
1. Jacobi, C. G. J. 1868, “Ueber die Aujl¨osung der Gleichung alx1 + a2x2 + . . . + anxn = fn”, J. Reine Angew. Math., vol. 69, pp. 1–28.
2. Jacobi, C. G. J. 1868, “Allgemeine Theorie der Kettenbruch¨ahnlichen Algorithmen, in welchen jede Zahl aus drei vorgegebenden gebildet wird”, J. Reine Angew. Math., vol. 69, pp. 29–64.
3. Perron, O. 1907, “Grundlagen f¨ur eine Theorie des Jacobischen Kettenbr¨uchalgorithmus”, Math. Ann., vol. 64, no. 1, pp. 1–76.
4. Bernstein, L 1971, The Jacobi–Perron algorithm. Its theory and application, Lecture notes in Mathematics, 207, Springer–Verlag, Berlin/Heidelberg/New York.
5. Bruin, M. G. de 1978, Convergence of generalized C-fractions // J. Approx. Theory, vol. 24, no. 3, pp. 177–207.
6. Bruin, M. G. de 1978, The interruption phenomenon for generalized continued fractions // Bull. Austral. Math. Soc., vol. 19, no. 2, pp. 245–272.
7. Poincar´e, H. 1885 “Sur les equations lin´eaires aux diff´erentielles et aux diff´erences finies // Amer. J. Math., vol. 7, pp. 203–258.
8. Perron, O. 1909, “ ¨Uber einen Satz des Herrn Poincare” // J. Reine Angew. Math., vol. 136, pp. 17–37.
9. Aptekarev, A. I. & Kalyagin, V. A. 1987, “Analytic properties of two-dimensional continued P-fraction expansions with periodic coefficients and their simultaneous Pade-Hermite approximants” // Lecture Notes in Math., vol. 1237, Springer-Verlag, pp. 145–160.
10. Nuttall, J. 1984, “Asymptotics of diagonal Hermite-Pad´e polynomials” // J. Approx. Theory, vol. 42, no. 4, pp. 299–386.
11. Nikishin, EM & Sorokin, VN 1988, Rational Approximations and Orthogonality, Nauka, Moscow.
12. Kalyagin, V. A. 1994, “Hermite–Pad´e approximants and spectral analysis of nonsymmetric operators” // Mat. Sb., vol. 185, no. 6, pp. 79–100.
13. Aptekarev, A. I. 1998, “Multiple Orthogonal Polynomials” // J. Comput. Appl. Math., vol. 99, no. 1–2, pp. 423–447
14. Aptekarev, A. I. & Kaliaguine, V. A. & Iseghem, J. van 2000 “The Genetic Sums’ Representation for the Moments of a System of Stieltjes Functions and its Application” // Constr. Approx., vol. 16, pp. 487–524.
15. Aptekarev, A. I. & Kaliaguine, V. A. & Saff, E. B. 2009 “Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials” // Constr. Approx., vol. 30, pp. 175–223.
16. Aptekarev, A. I. 2014, “Spectral Problems of High-Order Recurrences” // Amer. Math. Soc. Transl., vol. 233, pp. 43–61.
17. Lachaud, G. 1993, “Poly´edre d’Arnol’d et voile d’un cˆone simplicial: analogues du th´eorem de Lagrange” // C.R. Acad. Sci., Ser. 1. vol. 317. pp. 711–716.
18. Klein, F. 1895, “ ¨Uber eine geometrische Auffassung der gew¨ohnlichen Kettenbruchentwicklung” // Nachr. Ges. Wiss. G¨ottingen Math.-Phys. Kl., no. 3. pp. 357–359
19. Davenport, H. 1943, “On the product of three homogeneous linear form. IV” // Proc. Cambridge Philos. Soc., vol. 39. pp. 1–21.
20. Swinnerton-Dyer, H. P. F. 1971, “On the product of three homogeneous linear forms” // Acta Arithmetica, vol. 18. pp. 371–385.
21. Voronoi, GF 1896, On Generalization of the Algorithm of Continued Fraction, Warsawa University.
22. Bruno, A.D. 2003, “The correct generalization of the continued fraction”, Preprint no. 86 of the Keldysh Inst. of Applied Math., Moscow.
23. Bruno, A.D. 2006, “Generalizations of continued fraction”, Chebyshevsky sbornik, vol. 7, no. 3, pp. 4–71.
24. Bruno, A.D. 2015, “Universal generalization of the continued fraction algorithm”, Chebyshevsky sbornik, vol. 16, no. 2, pp. 35–65.
25. Bruno, A.D. 2016, “From Diophantine approximations to Diophantine equations”, Preprint no.1 of the Keldysh Inst. of Applied Math., Moscow.
Review
For citations:
Aptekarev A.I., Batkhin A.B., Bruno A.D. VLADIMIR IGOREVICH PARUSNIKOV. Chebyshevskii Sbornik. 2016;17(1):286-298. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-1-286-298
Views:
1577
Email this article 