( b ∗ δ$_{n}$, ψ ) converges to ( b, ψ ) . Since g, f belong to a convolution pair of spaces ( A, B ), we can enlarge the convolution pair ( A, B ) to include with every b ∈ A or B the corresponding sequences ˜ b$_{n}$. Denote the resulting pair of spaces by ( ˜ A, ˜ B ) . Show that this is also a convolution pair. Note that the convolution ˜ g$_{n}$ ∗ ˜ f$_{n}$ is defined as support of δ$_{n}$ is bounded. All that is needed is to show hypocontinuity; it follows from the fact that if a set T from A or B is bounded in S ∗ so is the corresponding set ˜ T that contains every element b ∈ T as well as all ˜ b$_{n}$. If a sequence b$_{m}$ converges to zero in S ∗ so does the corresponding ˜ b$_{mn}$, and hypocontinuity extends to the enlarged convolution pair. To show the exchange formula we need to establish that for any ( g, f ) ∈ ( A, B ) we get that F t ( g ) F t ( f ) = F t ( g ∗ f ) . Start by the exchange formula of Hirata and Ogata for the sequences: Consider first the left-hand side; by the continuity of Fourier transform in S ∗ lim F t (˜ g$_{n}$ ∗ ˜ f$_{n}$ ) = lim F t (˜ g$_{n}$ ) lim F t ( ˜ f$_{n}$ ) . lim F t (˜ g$_{n}$ ∗ ˜ f$_{n}$ ) = F t (lim(˜ g$_{n}$ ∗ ˜ f$_{n}$ )) , then by hypocontinuity of the convolution and because ˜ f$_{n}$ → f, ˜ g$_{n}$ → g we get that this is F t ( g ∗ f ) . 23