Walsh (2012) the solution can be given uniquely on W as long as at some point ζ$_{0}$$_{v}$ ∈ ( W$_{v}$ ∩ W ) the value β ( ζ$_{0}$$_{ν}$ ) is known for each of the connected components . Consider then β$_{1}$ ( ζ ) = Σ$_{ν}$[ β ( ζ$_{0}$$_{ν}$ ) exp ∫ ζ ζ$_{0}$ d ∑ k =1$^{κ}$ $_{k}$( ξ ) dξ ] I ( W$_{ν}$ ) , where integration is along any arc within the component that connects ζ to ζ$_{0}$$_{ν}$. Then α$_{1}$ = β − 1 1 γ, and α$_{2}$, β$_{2}$ are defined as above by being 1 on ∪$_{v}$ W$_{v}$ and arbitrary outside of this set. 

When β (0) = 1 as is the case for the characteristic function, the function is uniquely determined on the connected component that includes 0. 

Evdokimov and White (2012) provide a construction that permits in the univariate case to extend the solution β ( ζ$_{0}$$_{ν}$ ) [exp ∫ ζ ζ$_{0}$ d ∑ k =1$^{κ}$ $_{k}$( ξ ) dξ ] I ( W$_{ν}$ ) from a connected component of support where β ( ζ$_{0}$$_{ν}$ ) is known (e.g. at 0 for a characteristic function) to a contiguous connected component when on the border between the two where β = 0 , at least some finite order derivative of β is not zero. In the multivariate case this approach can be extended to the same construction along a one-dimensional arc from one connected compo- nent to the other. Thus identification is possible on a connected component of supp ( β, ∂ ) . 

Case 2. W is a closed set. 

Generally for models 5 and 7, W is the support of a generalized function and is a closed set. It may intersect with several connected components of support of β. Denote by W$_{v}$ here the intersection of a connected component of support of β and W. Then similarly β$_{1}$ ( ζ ) = ∑ ν [ β ( ζ$_{0}$$_{ν}$ ) exp ∫ ζ ζ$_{0}$ d ∑ k =1$^{κ}$ $_{k}$( ξ ) dξ ] I ( W$_{ν}$ ) , where integration is along any arc within the component that connects ζ to 

