Finite range decompositions of positive-definite functions |
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Authors: | David Brydges Anna Talarczyk |
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Institution: | a Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada b Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland |
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Abstract: | We give sufficient conditions for a positive-definite function to admit decomposition into a sum of positive-definite functions which are compactly supported within disks of increasing diameters Ln. More generally we consider positive-definite bilinear forms f→v(f,f) defined on . We say v has a finite range decomposition if v can be written as a sum v=∑Gn of positive-definite bilinear forms Gn such that Gn(f,g)=0 when the supports of the test functions f,g are separated by a distance greater or equal to Ln. We prove that such decompositions exist when v is dual to a bilinear form φ→∫2|Bφ| where B is a vector valued partial differential operator satisfying some regularity conditions. |
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Keywords: | Positive-definite Generalized Gaussian field Renormalisation group Elliptic operator Green's function |
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