Finite range decompositions of positive-definite functions

摘要

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 L-n. More generally we consider positive-definite bilinear forms f -> v(f, f) defined on C-0(infinity). We say v has a finite range decomposition if v can be written as a sum v = Sigma G(n) of positive-definite bilinear forms G(n) such that Gn (f, 9) = 0 when the supports of the test functions f, g are separated by a distance greater or equal to L-n. We prove that such decompositions exist when v is dual to a bilinear form phi -> vertical bar B phi vertical bar(2) where B is a vector valued partial differential operator satisfying some regularity conditions. (c) 2006 Elsevier Inc. All rights reserved.