Pointwise Kernels of Schwartz Distributions

We show that Schwartz distributions have kernels in the class of the pointwise nonstandard functions. The main purpose of this note is to show that every Schwartz distribution T G 3' has a kernel /: *Rd —> *C in the class of the pointwise nonstandard functions in the sense that (1) (T,p)= [ f(xy Y is a (standard) mapping, then */: *X —► *Y will be its nonstandard extension. We shall use the same notation, *, for the convolution operator *: 3' x 31 —> W and for its nonstandard extension *: *3' x *3 —» *l?. Lemma. There exists A in *3 such that for all q> in C° we have f A(x)*(x)dx = <p(<d). Received by the editors September 7, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 03H05, 30G10, 46F05, 46F10, 46F99.