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Pseudo-differential operators and equations in a discrete half-space

Abstract

We introduce a digital pseudo-differential operator acting in discrete Sobolev--Slobodetskii spaces and consider pseudo-differential equations with such operators in a discrete half-space.


The theorem on a general solution of such equations is proved for a special case.

Keyword : discrete functional space, digital distribution, digital pseudo-differential operator, discrete pseudo-differential equation, general solution

How to Cite
Vasilyev, V., & Vasilyev, A. (2018). Pseudo-differential operators and equations in a discrete half-space. Mathematical Modelling and Analysis, 23(3), 492-506. https://doi.org/10.3846/mma.2018.029
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Jul 4, 2018
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References

[1] A. Boettcher and B. Silbermann. Analysis of Toeplitz Operators. Springer-Verlag, Berlin, 2006.

[2] L. Botchway, P.G. Kibiti and M. Ruzhansky. Difference equations and pseudodifferential equations on z n . arXiv:1705.07564v1 [math.FA], pp. 1–29, 2017.

[3] G. Eskin. Boundary Value Problems for Elliptic Pseudodifferential Equations. AMS, Providence, 1981.

[4] L. Frank. Spaces of network functions. Math. USSR Sb., 15(2):182–226, 1971. https://doi.org/10.1070/SM1971v015n02ABEH001541.

[5] F.D. Gakhov. Boundary Value Problems. Dover Publications, Mineola, 1981.

[6] I.C. Gohberg and I.A. Feldman. Convolution Equations and Projection Methods for Their Solution. AMS, Providence, 1974.

[7] R. Hagen, S. Roch and B. Silbermann. C ∗ -algebras and Numerical Analysis. Marcel Dekker, New York, 2001.

[8] I. Lifanov, L. Poltavskii and G. Vainikko. Hypersingular Integral Equations and their Applications. Chapman & Hall/CRC, Boca Raton, 2004.

[9] I.K. Lifanov. Singular Integral Equations and Discrete Vortices. VSP, Utrecht, 1996.

[10] I.K. Lifanov and L.N. Poltavskii. Pseudodifference operators and uniform convergence of divided differences. Sbornik: Mathematics, 193(2):205–230, 2002. https://doi.org/10.1070/SM1971v015n02ABEH001541.

[11] S.G. Mikhlin and S. Proessdorf. Singular Integral Operators. Springer-Verlag, Berlin, 1986.

[12] N.I. Muskhelishvili. Singular Integral Equations. North Holland, Amsterdam, 1976.

[13] S. Proessdorf and B. Silbermann. Numerical Analysis for Integral and Related Operator Equations. Birkhauser, Basel, 1991.

[14] V. Rabinovich. Wiener algebra of operators on the lattice µz n depending on the small parameter µ > 0. Complex Var. Ell. Equ., 58(6):751–766, 2013. https://doi.org/10.1070/SM1971v015n02ABEH001541.

[15] J. Saranen and G. Vainikko. Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer, Berlin, 2002. https://doi.org/10.1070/SM1971v015n02ABEH001541.

[16] M. Taylor. Pseudodifferential Operators. Princeton University Press, Princeton, 1981.

[17] F. Treves. Introduction to Pseudodifferential Operators and Fourier Integral Operators. Springer, New York, 1980. https://doi.org/10.1070/SM1971v015n02ABEH001541.

[18] G. Vainikko. Multidimensional Weakly Singular Integral Equations. SpringerVerlag, Berlin – Heidelberg, 1993. https://doi.org/10.1007/BFb0088979.

[19] A. Vasil’ev and V. Vasil’ev. On the solvability of certain discrete equations and related estimates of discrete operators. Doklady Math., 92(2):585–589, 2015. https://doi.org/10.1007/BFb0088979.

[20] A. V. Vasil’ev and V. B. Vasil’ev. Periodic Riemann problem and discrete convolution equations. Differ. Equ., 51(5):652–660, 2015. https://doi.org/10.1134/S0012266115050080.

[21] A. Vasilyev and V. Vasilyev. Numerical analysis for some singular integral equations. Neural Parallel Sci. Comput., 20(3):313–326, 2012.

[22] A. Vasilyev and V. Vasilyev. Discrete singular operators and equations in a half-space. Azerb. J. Math., 3(1):84–93, 2013.

[23] A.V. Vasilyev and V.B. Vasilyev. Discrete singular integrals in a half-space. In V. Mityushev and M. Ruzhansky(Eds.), Current Trends in Analysis and Its Applications, Trends in Mathematics. Research Perspectives, pp. 663–670, Basel, 2015. Birkhauser. https://doi.org/10.1134/S0012266115050080.

[24] V. Vasilyev. Discrete equations and periodic wave factorization. In A. Ashyralyev and A. Lukashov(Eds.), Proceedings of the International Conference on Analysis and Applied Mathematics (ICAAM-2016), volume 1759 of AIP Conf. Proc., p. 020126, Melville, 2016. AIP Publishing. https://doi.org/10.1063/1.4959740.

[25] V. Vasilyev. Discreteness, periodicity, holomorphy, and factorization. In C. Constanda, M. Dalla Riva, P.D. Lamberti and P. Musolino(Eds.), Integral Methods in Science and Engineering, volume 1 of Theoretical Technique, pp. 315–324, New York, 2017. Birkhauser. https://doi.org/10.1063/1.4959740.

[26] V. Vasilyev. On discrete boundary value problems. In T. Kal’menov and M. Sadybekov(Eds.), Proceedings of the International Conference “Functional Analysis in Interdisciplinary Applications” (FAIA2017), volume 1880 of AIP Conf. Proc., p. 050010, Melville, 2017. AIP Publishing. https://doi.org/10.1063/1.4959740.

[27] V. Vasilyev. The periodic Cauchy kernel, the periodic Bochner kernel, discrete pseudo-differential operators. In T. Simos and C. Tsitouras(Eds.), Proceedings of the International Conference on Numerical Analysis and Applications (ICNAAM-2016), volume 1863 of AIP Conf. Proc., p. 140014, Melville, 2017. AIP Publishing. https://doi.org/10.1063/1.4959740.

[28] V.S. Vladimirov. Generalized Functions in Mathematical Physics. Mir, Moscow, 1979.