Isomorphism between one-dimensional and multidimensional finite difference operators

Anton A. Kutsenko

doi:10.3934/cpaa.2020270

Article Contents

2021, Volume 20, Issue 1: 359-368. Doi: 10.3934/cpaa.2020270

This issue Previous Article Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound Next Article On optimal autocorrelation inequalities on the real line

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

Early access: November 2020

Published: January 2021

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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Abstract

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:

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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} $

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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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