g \ f D S E ∗ O ∗ C S ∗ D ∗ D D S D S O$_{M}$ C$_{∞}$ S S S S S O$_{M}$ X E ∗ D S E ∗ O ∗ C S ∗ D ∗ O ∗ C S S O ∗ C O ∗ C S ∗ X S ∗ O$_{M}$ O$_{M}$ S ∗ S ∗ X X D ∗ C$_{∞}$ X D ∗ X X X Table 1: The convolution table pairs of spaces for elements of which convolution is defined (X indicates that convolution cannot be defined for some pairs of elements of the spaces); the table entries indicate to which space the element resulting from the convo- lution operation belongs. The table is an extended version of the one in the textbook by Kirillov and Gvishiani (1982, p.102) and summarizes the well-established results in the literature. The convolution pairs in the table where convolution is defined all possess the hypocontinuity property (Schwartz, 1966, p.167, p.247-257). Hypocon- tinuity of a bilinear operation means that if one component of a pair is in a bounded set in G ∗ and the other converges to zero in G $^{∗}$, the result of the bilinear operation converges to zero (Schwartz, 1966, pp.72,73). Convolution of a pair of arbitrary generalized functions is not always defined in S $^{∗}$. Bounded support or at least rapid decline at infinity is needed for a convolution to exist. For example, convolution of a constant function with another constant function on R 1 is not defined. Nevertheless there are pairs of subspaces of generalized functions beyond those in the Table for which convolution defines a generalized function; spaces where convolution is 19