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ESTIMATES FOR SOLUTIONS TO DISCRETE CONVOLUTION EQUATIONS

Part of: Inversion theorems Equations and systems with constant coefficients General convexity Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 April 2015

Christer O. Kiselman
Christer O. Kiselman*
Affiliation:
Department of Information Technology, Division of Visual Information and Interaction, Computerized Image Analysis and Human–Computer Interaction, Uppsala University, PO Box 337, SE-751 05 Uppsala, Sweden email [email protected], [email protected]
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Abstract

We study solvability of convolution equations for functions with discrete support in $\mathbf{R}^{n}$, a special case being functions with support in the integer points. The more general case is of interest for several grids in Euclidean space, like the body-centred and face-centred tessellations of 3-space, as well as for the non-periodic grids that appear in the study of quasicrystals. The theorem of existence of fundamental solutions by de Boor et al is generalized to general discrete supports, using only elementary methods. We also study the asymptotic growth of sequences and arrays using the Fenchel transformation.

MSC classification

Primary: 40E10: Growth estimates 65M06: Finite difference methods
Secondary: 35E05: Fundamental solutions 35E15: Initial value problems 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 295 - 308
Copyright
Copyright © University College London 2015 

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References

de Boor, C., Höllig, K. and Riemenschneider, S., Fundamental solutions for multivariate difference equations. Amer. J. Math. 111(3) 1989, 403–415.CrossRefGoogle Scholar
Bourbaki, N., Théorie des ensembles (Éléments de mathématique, Première partie), Hermann (Paris, 1954), Chs. I and II.Google Scholar
Bourbaki, N., Topologie générale (Éléments de mathématique, Première partie), 3rd edn., Hermann (Paris, 1961), Chs. 1 and 2.Google Scholar
Domar, Y., Convolution theorems of Titchmarsh type on discrete Rn. Proc. Edinb. Math. Soc. (2) 32(3) 1989, 449–457.CrossRefGoogle Scholar
Hörmander, L., On the division of distributions by polynomials. Ark. Mat. 3 1958, 555–568.CrossRefGoogle Scholar
Łojasiewicz, S., Division d’une distribution par une fonction analytique de variables réelles. C. R. Acad. Sci. Paris 246 1958, 683–686.Google Scholar
Łojasiewicz, S., Sur le problème de la division. Studia Math. 18 1959, 87–136.CrossRefGoogle Scholar
Strand, R., Distance Functions and Image Processing on Point-Lattices. With Focus on the 3D Face- and Body-centered Cubic Grids (Uppsala Dissertations from the Faculty of Science and Technology 79), Acta Universitatis Upsaliensis (Uppsala, 2008).Google Scholar
Weiss, B., Titchmarsh’s convolution theorem on groups. Proc. Amer. Math. Soc. 19 1968, 75–79.Google Scholar