# American Institute of Mathematical Sciences

January  2021, 20(1): 359-368. doi: 10.3934/cpaa.2020270

## Isomorphism between one-dimensional and multidimensional finite difference operators

 Jacobs University, Campus Ring 1, 28759 Bremen, Germany, Saint-Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia

Received  July 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: This paper is a contribution to the project M3 of the Collaborative Research Centre TRR 181 "Energy Transfer in Atmosphere and Ocean" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 274762653. This work is also supported by the RFBR (RFFI) grant No. 19-01-00094

Finite difference operators are widely used for the approximation of continuous ones. It is well known that the analysis of continuous differential operators may strongly depend on their dimensions. We will show that the finite difference operators generate the same algebra, regardless of their dimension.

Citation: Anton A. Kutsenko. Isomorphism between one-dimensional and multidimensional finite difference operators. Communications on Pure and Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270
##### References:
 [1] V. V. Bavula, The algebra of integro-differential operators on an affine line and its modules, J. Pure Appl. Algebra, 217 (2013), 495-529.  doi: 10.1016/j.jpaa.2012.06.024. [2] K. Davidson, C*-Algebras by Example, American Mathematical Society and Fields Institute, 1997. doi: 10.1090/fim/006. [3] L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, Rhode Island, 2010. [4] J. G. Glimm, On a certain class of operator algebras, Trans. Am. Math. Soc., 95 (1960), 318-340.  doi: 10.2307/1993294. [5] L. Guo and W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra, 212 (2008), 522-540.  doi: 10.1016/j.jpaa.2007.06.008. [6] L. Guo, G. Regensburger and M. Rosenkranz, On integro-differential algebras, J. Pure Appl. Algebra, 218 (2014), 456-473.  doi: 10.1016/j.jpaa.2013.06.015. [7] R. Hoegh-Krohn and T. Skjelbred, Classification of C*-algebras admitting ergodic actions of the two-dimensional torus, J. Reine Angew. Math., 328 (1981), 1-8.  doi: 10.1515/crll.1981.328.1. [8] M. Rordam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000. [9] M. Rosenkranz, A new symbolic method for solving linear two-point boundary value problems on the level of operators, J. Symb. Comput, 39 (2005), 171-199.  doi: 10.1016/j.jsc.2004.09.004. [10] M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover Publications Inc., 2012. [11] H. S. Yin, A simple proof of the classification of rational rotation C*-algebras, P. Am. Math. Soc., 98 (1986), 469-470.  doi: 10.2307/2046204.

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##### References:
 [1] V. V. Bavula, The algebra of integro-differential operators on an affine line and its modules, J. Pure Appl. Algebra, 217 (2013), 495-529.  doi: 10.1016/j.jpaa.2012.06.024. [2] K. Davidson, C*-Algebras by Example, American Mathematical Society and Fields Institute, 1997. doi: 10.1090/fim/006. [3] L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, Rhode Island, 2010. [4] J. G. Glimm, On a certain class of operator algebras, Trans. Am. Math. Soc., 95 (1960), 318-340.  doi: 10.2307/1993294. [5] L. Guo and W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra, 212 (2008), 522-540.  doi: 10.1016/j.jpaa.2007.06.008. [6] L. Guo, G. Regensburger and M. Rosenkranz, On integro-differential algebras, J. Pure Appl. Algebra, 218 (2014), 456-473.  doi: 10.1016/j.jpaa.2013.06.015. [7] R. Hoegh-Krohn and T. Skjelbred, Classification of C*-algebras admitting ergodic actions of the two-dimensional torus, J. Reine Angew. Math., 328 (1981), 1-8.  doi: 10.1515/crll.1981.328.1. [8] M. Rordam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000. [9] M. Rosenkranz, A new symbolic method for solving linear two-point boundary value problems on the level of operators, J. Symb. Comput, 39 (2005), 171-199.  doi: 10.1016/j.jsc.2004.09.004. [10] M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover Publications Inc., 2012. [11] H. S. Yin, A simple proof of the classification of rational rotation C*-algebras, P. Am. Math. Soc., 98 (1986), 469-470.  doi: 10.2307/2046204.
Two first partitions for the unitary transform ${\mathcal U}_{2,1}^{-1}$ between $L^2_{2,1}$ and $L^2_{1,1}$ are shown. The characteristic functions of squares and intervals with the same "blue" and "red" numbers are transformed into each other under the action of ${\mathcal U}_{2,1}$
The unitary transform ${\mathcal U}_{2,1}^{-1}$, see Fig. 1, applied to the function $z(x,y) = 1+\sin(\pi(x^2+y^2))$
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