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recurrence relations; linear difference equations; linear multidimensional digital systems; nonlinear partial difference equations
In this paper linear difference equations with several independent variables are considered, whose solutions are functions defined on sets of $n$-dimensional vectors with integer coordinates. These equations could be called partial difference equations. Existence and uniqueness theorems for these equations are formulated and proved, and interconnections of such results with the theory of linear multidimensional digital systems are investigated. Numerous examples show essential differences of the results from those of the theory of (one-dimensional) difference equations. The significance and the correct formulation of initial conditions for the solution o partial difference equation is established and methods are described, which make it possible to construct the solution algorithmically. Extensions of the theory to some special nonlinear partial difference equations are also considered.
[1] P. S. Alexandrov: Introduction to general theory of sets and functions. (Russian) Czech translation, Academia Praha, 1954.
[2] N. K. Bose: Applied Multidimensional Systems Theory. Van Nostrand, N.Y. 1982. MR 0652483 | Zbl 0574.93031
[3] D. E. Dudgeon R. M. Mercereau: Multidimensional Digital Signal Processing. Prentice-Hall, Engelwood Cliffs, 1984.
[4] A. O. Gelfond: Finite difference calculus. (in Russian). Nauka, Moscow, 1967. MR 0216186
[5] T. S. Huang (Editor): Two Dimensional Digital Signal Processing. Vol. I., Springer, Berlin, 1981.
[6] B. Pondělíček: On compositional and convolutional systems. Kybernetika (Prague), 17 (1981), No 4, pp. 277-286. MR 0643915
[7] S. L. Sobolev: Introduction to theory of cubature formulae. (in Russian). Nauka, Moscow, 1974. MR 0478560
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