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

Isomorphism between one-dimensional and multidimensional finite difference operators

Anton A. Kutsenko

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.

Keywords: Representation of finite difference operators, UHF algebras, ODE and PDE.
Mathematics Subject Classification: Primary: 35P99, 16G99; Secondary: 39A05, 39A14.
Citation: Anton A. Kutsenko. Isomorphism between one-dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270
References:
[1]

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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.  Google Scholar

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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.  Google Scholar

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M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover Publications Inc., 2012. Google Scholar

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show all references

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.  Google Scholar

[2]

K. Davidson, C*-Algebras by Example, American Mathematical Society and Fields Institute, 1997. doi: 10.1090/fim/006.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, Rhode Island, 2010. Google Scholar

[4]

J. G. Glimm, On a certain class of operator algebras, Trans. Am. Math. Soc., 95 (1960), 318-340.  doi: 10.2307/1993294.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8] M. Rordam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000.   Google Scholar
[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.  Google Scholar

[10]

M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover Publications Inc., 2012. Google Scholar

[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.  Google Scholar

$ {\mathcal U}_{2,1} $">Figure 1.  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} $
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Fig. 1, applied to the function $ z(x,y) = 1+\sin(\pi(x^2+y^2)) $">Figure 2.  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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