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On the right hand side by continuity of Fourier transform and convergence 

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Denote the space of Fourier transforms of elements from A (or B ) by F t ( A ) (correspondingly, F t ( B )) . Then ( F t ( A ) , F t ( B )) is a product pair. Hypocon- tinuity of the product follows immediately from the hypocontinuity of the convolution and continuity of the Fourier transform and its inverse. 

## Corollary 1. Given a product pair of spaces (Γ , Φ) in S ∗ the exchange formula applies to any γ, φ ∈ (Γ , Φ) : 

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The proof follows the proof of Theorem 1 above by replacing convolution with product and Fourier transform by the inverse Fourier transform. 

In examining problems such as deconvolution much of the literature fo- cuses on Fourier transforms, e.g. characteristic functions. The dichotomy between convolution pairs of spaces and product pairs of spaces in S ∗ allows to switch between the two types of pairs. 

If φ is the characteristic function of a measurement or contamination error the condition φ ∈ O$_{M}$ for the classical product pair it would require existence of all moments. It may be of interest to consider pairs where products of Fourier transforms of generalized functions exist for less smooth functions (e.g. with relaxed moments requirements on measurement error); 

