Degenerate Scalar Invariants and the Groups of Motion of a Riemann Space.

Using Theorem 1 (or, rather, the analogous result for higher-dimensional Euclidean spaces) we deduce4' 6 THEOREM 2. Let D be any partial differential operator with constant coefficients, and let T be any distribution. Then we can find a distribution S such that DS = T. This result is also valid if D is a partial differential-difference operator with constant coefficients. Remark: We can also give a direct description of the topology of the space E' which is the Fourier transform of the space of distributions of compact carrier.2' This leads to another proof of the result6 that, if D is a partial differential-difference operator with constant coefficients, and f is an indefinitely differentiable function, than we can find an indefinitely differentiable function g such that Dg = f. By similar methods we can also show that, if f is any entire function, there exists an entire function g with Dg = f.