# On generalized difference equations

Aplikace matematiky (1987)

• Volume: 32, Issue: 3, page 224-239
• ISSN: 0862-7940

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## Abstract

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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.

## How to cite

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Bosák, Miroslav, and Gregor, Jiří. "On generalized difference equations." Aplikace matematiky 32.3 (1987): 224-239. <http://eudml.org/doc/15495>.

@article{Bosák1987,
abstract = {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.},
author = {Bosák, Miroslav, Gregor, Jiří},
journal = {Aplikace matematiky},
keywords = {recurrence relations; linear difference equations; linear multidimensional digital systems; nonlinear partial difference equations; recurrence relations; linear difference equations; linear multidimensional digital systems; nonlinear partial difference equations},
language = {eng},
number = {3},
pages = {224-239},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On generalized difference equations},
url = {http://eudml.org/doc/15495},
volume = {32},
year = {1987},
}

TY - JOUR
AU - Bosák, Miroslav
AU - Gregor, Jiří
TI - On generalized difference equations
JO - Aplikace matematiky
PY - 1987
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 32
IS - 3
SP - 224
EP - 239
AB - 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.
LA - eng
KW - recurrence relations; linear difference equations; linear multidimensional digital systems; nonlinear partial difference equations; recurrence relations; linear difference equations; linear multidimensional digital systems; nonlinear partial difference equations
UR - http://eudml.org/doc/15495
ER -

## References

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1. P. S. Alexandrov, Introduction to general theory of sets and functions, (Russian) Czech translation, Academia Praha, 1954. (1954)
2. N. K. Bose, Applied Multidimensional Systems Theory, Van Nostrand, N.Y. 1982. (1982) Zbl0574.93031MR0652483
3. D. E. Dudgeon R. M. Mercereau, Multidimensional Digital Signal Processing, Prentice-Hall, Engelwood Cliffs, 1984. (1984) Zbl0643.94001
4. A. O. Gelfond, Finite difference calculus, (in Russian). Nauka, Moscow, 1967. (1967) MR0216186
5. T. S. Huang (Editor), Two Dimensional Digital Signal Processing, Vol. I., Springer, Berlin, 1981. (1981)
6. B. Pondělíček, On compositional and convolutional systems, Kybernetika (Prague), 17 (1981), No 4, pp. 277-286. (1981) Zbl0469.93066MR0643915
7. S. L. Sobolev, Introduction to theory of cubature formulae, (in Russian). Nauka, Moscow, 1974. (1974) MR0478560

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