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relaxing the smoothness of φ will require restricting the degree of singularity of γ. 

For a continuous function φ ∈ C (0) define the space of test functions, denoted G ⊕ φG, that consists of functions that can be represented as ψ$_{1}$ + φψ$_{2}$, with ψ$_{i}$ ∈ G ( G = D or S ). Consider γ ∈ G ∗ that can be extended to a continuous linear functional on G ⊕ φG, denote the linear space generated by such γ by G ( φ ) $^{∗}$. If φ ∈ G then G ( φ ) ∗ = G $^{∗}$. For any ψ ∈ G the value ( γ, φψ ) is defined and is a continuous functional with respect to ψ ; this defines the generalized function φγ : ( φγ, ψ ) = ( γ, φψ ) . Then ( G ( φ ) ∗ , G ⊕ φG ) is a product pair; the corresponding spaces of inverse Fourier transforms form a convolution pair. 

For example, the derivative, δ $^{′}$, of a univariate Dirac δ − function in G ∗ can be multiplied by any continuous function, φ, that is differentiable at 0, since then ( δ $^{′}$φ, ψ ) = ( δ $^{′}$, φψ ) with ( δ $^{′}$, φψ ) = φ $^{′}$(0) ψ (0) + φ (0) ψ $^{′}$(0) . More generally, for a product between a continuous function and a generalized function to be defined, there is a trade-off between differentiability of the continuous function and the degree of singularity of the generalized function. 

Under the Assumptions 1-3 the convolution equations (2,3) lead to cor- responding equations for Fourier transforms: 

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