On Correctness of Cauchy problem
for a Polynomial Difference Operator
with Constant Coefficients
The theory of linear difference equations is applied in various areas of mathematics and in the one-dimensional case is quite established. For n > 1, the situation is much more difficult and even for the constant coefficients a general description of the space of solutions of a difference equation is not available.
In the combinatorial analysis, difference equations combined with the method of generating functions produce a powerful tool for investigation of enumeration problems. Another instance when difference equations appear is the discretization of differential equations. In particular, the discretization of the Cauchy–Riemann equation led to the creation of the theory of discrete analytic functions which found applications in the theory of Riemann surfaces and the combinatorial analysis. The methods of discretization of a differential problem are an important part of the theory of difference schemes and also lead to difference equations. The existence and uniqueness of a solution is one of the main questions in the theory of difference schemes.
Another important question is the stability of a difference equation. For n = 1 and constant coefficients the stability is investigated in the framework of the theory of discrete dynamical systems and is completely defined by the roots of the characteristic polynomial, namely: they all lie in the unit disk.
In the present work, we give two easily verified sufficient conditions on the coefficients of a difference operator which guarantee the correctness of a Cauchy problem.
Marina S. Apanovich, Cand. Sci. (Phys.–Math.), Krasnoyarsk State Medical University named after Prof. V. F. Voino-Yasenetsky, 1, Partizan Zheleznyak st., Krasnoyarsk, 660022, Russian Federation, e-mail: email@example.com
Evgeny K. Leinartas, Dr. Sci. (Phys.–Math.), Prof., Institute of Mathematics and Computer Science, Siberian Federal University, 79, Svobodny pr., Krasnoyarsk, 660041, Russian Federation, e-mail: firstname.lastname@example.org
Apanovich M.S., Leinartas E.K. On Correctness of Cauchy problem for a Polynomial Difference Operator with Constant Coefficients. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 26, pp. 3-15. https://doi.org/10.26516/1997-7670.2018.26.3
- Dadzhion D., Mersero R. Tsifrovaya obrabotka mnogomernykh signalov [Digital processing of multidimensional signals]. Moscow, Mir Publ., 1988, 488 p. (in Russian)
- Leinartas E.K. Multiple Laurent series and fundamental solutions of linear difference equations. Siberian Mathematical Journal, 2007, vol. 48, no. 2, pp. 268-272. https://doi.org/10.1007/s11202-007-0026-0
- Leinartas E.K., Lyapin A.P. O ratsional’nosti mnogomernykh vozvratnykh stepennykh ryadov [On rationality multidimentional recursive power series]. Journal of Siberian Federal University, 2009, vol. 2, no. 4, pp. 449-455. (in Russian)
- Leynartas E.K. Stability of the Cauchy problem for a multidimensional difference operator and the amoeba of the characteristic set. Siberian Mathematical Journal, 2011, vol. 52, no. 5, pp. 864-870. https://doi.org/10.1134/S0037446611050119
- Leynartas E.K., Rogozina M.S. Solvability of the Cauchy problem for a polynomial difference operator and monomial bases for the quotients of a polynomial ring. Siberian Mathematical Journal, 2015, vol. 56, no. 1, pp. 92-100. https://doi.org/10.1134/S0037446615010097
- Nekrasova T.I. Ob ierarkhii proizvodyashchikh funktsiy resheniy mnogomernykh raznostnykh uravneniy [On the Hierarchy of Generating Functions for Solutions of Multidimensional Difference Equations]. The Bulletin of Irkutsk State University. Series Mathematics, 2014, vol. 9, pp. 91-102. (in Russian)
- Rogozina M.S. O razreshimosti zadachi Koshi dlya polinomial’nogo raznostnogo operatora [On the solvability of the Cauchy problem for a polynomial difference operator]. Vestnik NGU. Serija: Matematika, mehanika, informatika [Bulletin of NSU. Series: Mathematics, Mechanics, Informatics], 2014, vol. 14, no. 3, pp. 83–94.
- Rjaben’kiy V.S., Filippov A.F. Ob ustoychivosti raznostnykh uravneniy [On the stability of difference equations]. Moscow, Gosudarstvennoe izdatel’stvo tekhnikoteoreticheskoy literatury, 1956, 174 p. (in Russian)
- Samarskiy A.A. Teoriya raznostnykh skhem [Theory of difference schemes]. Moscow, Nauka Publ., 1977, 656 p. (in Russian)
- Fedoryuk M.V. Asimptotika: integraly i ryady [Asymptotics. Integrals and series]. Moscow, Nauka Publ., 1987, 546 p.
- Hormander L. Lineynye differentsial’nye operatory s chastnymi proizvodnymi [Linear Differential Operators with Partial Derivatives]. Moscow, Mir Publ., 1965. 379 p. (in Russian)
- Hormander L. Vvedenie v teoriyu funktsiy neskol’kikh kompleksnykh peremennykh [An introduction to complex analysis in several variables]. Moscow, Mir Publ., 1968, 280 p. (in Russian)
- Tsikh A.K. Conditions for absolute convergence of the taylor coefficient series of a meromorphic function of two variables. Mathematics of the USSR-Sbornik, 1993, vol. 74, no. 2, pp. 337-360. http://dx.doi.org/10.1070/SM1993v074n02ABEH003350
- Shabat B.V. Vvedenie v kompleksnyy analiz. Funktsii odnogo peremennogo [An introduction to complex analysis. Functions of one variable]. Moscow, Lenand Publ., 2015, 336 p. (in Russian)
- Shabat B.V. Vvedenie v kompleksnyy analiz. Funktsii neskol’kikh peremennykh [An introduction to complex analysis. Functions of several variables]. Moscow, Lenand Publ., 2015, 464 p. (in Russian)
- Ahlberg J.H., Nilson E.N. Convergence properties of the spline fit. J.SIAM, 1963, vol.11, no. 1, pp. 95-104. https://doi.org/10.1137/0111007
- Apanovich M.S., Leinartas E.K. Correctness of a Two-dimensional Cauchy Problem for a Polynomial Difference Operator with Constant Coefficients. Journal of Siberian Federal University. Mathematics & Physics, 2017, vol.10, no. 2, pp. 199-205. https://doi.org/10.17516/1997-1397-2017-10-2-199-205
- Bousquet-Melou M., Petkovˇsek M. Linear recurrences with constant coefficients: the multivariate case. Discrete Mathematics, 2000, vol. 225, pp. 51-75. https://doi.org/10.1016/S0012-365X(00)00147-3
- Taussky O. A recurring theorem on determinants. The American Mathematical Monthly, 1949, vol. 56, no. 10, pp. 672-676. https://doi.org/10.1080/00029890.1949.11990209