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Isomorphism between one-dimensional and multidimensional finite difference operators

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

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  • 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.

    Mathematics Subject Classification: Primary: 35P99, 16G99; Secondary: 39A05, 39A14.

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  • 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} $

    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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