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of solving these convolution equations in spaces of generalized functions: existence of a unique solution (identification) and continuity of the mapping from the known functions to the solution (well-posedness). 

The usual blueprint for deconvolution in function spaces works as follows. Assume: g ∗ f = w holds, convolution exists, e.g. for functions in L$_{1}$, say, densities. Fourier transform ( F t ) is defined: for the function g, F t ( g ) = ∫ g ( x ) exp( ix $^{T}$ζ ) dx ; this is a characteristic function if g is a density. Exchange formula applies: for Fourier transforms γ = F t ( g ); φ = F t ( f ); ε = F t ( w ) a convolution is transformed into product: 

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Also, if additionally Fourier transform exists for h$_{k}$ ( x ) = x$_{k}$g ( x ) , then F t ( h$_{k}$ ) = − i ∂ $_{∂ξ$_{k}$}$γ ( ξ ); denote the derivative ∂ $_{∂ξ$_{k}$}$γ ( ξ ) by γ ′ $_{k}$. 

If w and f in equation (2) (thus also ε and φ ) are known and φ ̸ = 0 solve the algebraic equation : γ = φ − $^{1}$ε, then apply the inverse Fourier transform, F t − $^{1}$, to obtain g, 

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When the functions that enter the equations are estimated based on avail- able data on the observables, the solutions will be stochastic and for estab- lishing consistency well-posedness of the solutions becomes crucial; Carrasco et al (2007), An and Hu (2012) discuss well-posedness that applies to similar problems in various normed spaces, mostly in spaces of integrable functions, 

