Geodesic
Difference and discrete equations on the real axis and the semiaxis
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 46 (2015) no. 2, pp. 29-37.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider general difference equations on the real axis and the semiaxis in terms of the theory of pseudo-differential equations and boundary value problems. It is shown that a principal role for describing the picture of solvability for such equations plays an index of factorization of an elliptic symbol of difference equation with constant coefficients. A form of solution of such equation in the spaces of square integrable functions is described and necessary and sufficient conditions for solvability are obtained. Some discrete analogues of such equations are described.
Keywords: difference equation, discrete equation, index of factorization, solvability, general solution.
Mots-clés : symbol 
@article{IIMI_2015_46_2_a3,
     author = {A. V. Vasil'ev and V. B. Vasil'ev},
     title = {Difference and discrete equations on the real axis and the semiaxis},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {29--37},
     publisher = {mathdoc},
     volume = {46},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2015_46_2_a3/}
}
TY  - JOUR
AU  - A. V. Vasil'ev
AU  - V. B. Vasil'ev
TI  - Difference and discrete equations on the real axis and the semiaxis
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2015
SP  - 29
EP  - 37
VL  - 46
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IIMI_2015_46_2_a3/
LA  - ru
ID  - IIMI_2015_46_2_a3
ER  - 
%0 Journal Article
%A A. V. Vasil'ev
%A V. B. Vasil'ev
%T Difference and discrete equations on the real axis and the semiaxis
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2015
%P 29-37
%V 46
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IIMI_2015_46_2_a3/
%G ru
%F IIMI_2015_46_2_a3
A. V. Vasil'ev; V. B. Vasil'ev. Difference and discrete equations on the real axis and the semiaxis. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 46 (2015) no. 2, pp. 29-37. http://geodesic.mathdoc.fr/item/IIMI_2015_46_2_a3/

[1] Gakhov F. D., Boundary value problems, Nauka, M., 1977, 640 pp.

[2] Muskhelishvili N. I., Singular integral equations, Nauka, M., 1968, 511 pp.

[3] Mikhlin S. G., Prößdorf S., Singular integral operators, Springer, Berlin–Heidelberg–New York, 1986, 528 pp. | MR | Zbl

[4] Simonenko I. B., Local method in theory of operators that are invariant under a shift and their envelopes, TsVVR, Rostov-on-Don, 2007, 120 pp.

[5] Gokhberg I. Ts., Krupnik N. Ya., Introduction to theory of one-dimensional singular integral equations, Shtiintsa, Chisinau, 1973, 426 pp.

[6] Eskin G. I., Boundary value problems for elliptic pseudodifferential equations, Translations of Mathematical Monographs, 52, AMS, 1981, 375 pp. | MR | Zbl

[7] Vasil'ev V. B., Fourier integral multipliers, pseudodifferential equations, wave factorization, and boundary value problems, Editorial URSS, M., 2010, 135 pp.

[8] Titchmarsh E., Introduction to the theory of Fourier integrals, The Clarendon press, Oxford, 1937, 390 pp. | Zbl

[9] Milne-Thomson L. M., The calculus of finite differences, Chelsea Publishing Company, New York, 1981, 558 pp. | MR | Zbl

[10] Jordan C., Calculus of finite differences, Chelsea Publishing Company, New York, 1965, 654 pp. | MR | Zbl

[11] Vasilyev V. B., “General boundary value problems for pseudo differential equations and related difference equations”, Advances in Difference Equations, 2013:289 (2013), 1–7 | DOI | MR

[12] Vasilyev V. B., “On some difference equations of first order”, Tatra Mt. Math. Publ., 54 (2013), 165–181 | MR | Zbl

[13] Vasilyev A. V., Vasilyev V. B., “Discrete singular operators and equations in a half-space”, Azerbaijan Journal of Mathematics, 3:1 (2013), 84–93 | MR | Zbl

[14] Vasilyev A. V., Vasilyev V. B., “Discrete singular integrals in a half-space”, Current Trends in Analysis and its Applications, Proceedings of the 9th ISAAC Congress (Krakow, Poland, 2013), 663–670 | Zbl

[15] Vasil'ev A. V., Vasil'ev V. B., “Periodic Riemann problem and discrete convolution equations”, Differential Equations, 51:5 (2015), 652–660 | DOI | DOI | Zbl

[16] Sobolev S. L., Introduction to the theory of cubature formulae, Nauka, M., 1974, 808 pp.

[17] Dudgeon D., Mersereau R., Multidimensional digital signal processing, Prentice Hall, Englewood Cliffs, New Jersey, 1984, 400 pp. | Zbl

[18] Kurbatov V. G., Linear differential-difference equations, Voronezh University Publ., Voronezh, 1990, 167 pp.

[19] Kamenskii G. A., Skubachevskii A. L., Linear boundary-value problems for differential-difference equations, MAI, M., 1992, 190 pp.

[20] Noble B., Methods based on the Wiener–Hopf technique for the solution of partial differential equations, Pergamon Press, London–New York–Paris–Los Angeles, 1958, 246 pp. | MR | Zbl