Finite range decompositions of positive-definite functions

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 C∞0 . 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→ |Bϕ|2 where B is a vector valued partial differential operator satisfying some regularity conditions. © 2006 Elsevier Inc. All rights reserved.