Bull. Amer. Math. Soc., **1**(77), p. 120-123

DOI: 10.1090/s0002-9904-1971-12627-7

Elsevier, *Journal of Functional Analysis*, **1**(8), p. 173-188

DOI: 10.1016/0022-1236(71)90025-5

Journal article published in 1971 by
Gary H. Meisters

This paper is made freely available by the publisher.

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archiving forbidden

It is shown in this paper (Theorem 1) that if α and β are real numbers such that is irrational and algebraic, then there exist two (necessarily distinct) distributions S and T on R, both with compact supports, such that δ′ = ΔαS + ΔβT. Here ΔαS means S − Sα and Sα denotes the translate of S by α. It is also shown that δ′ has no such representation if has rational or certain transcendental values, and that S and T can be chosen to have order two and no lower order. If ϑ belongs to any of the spaces or their duals then belong to the same space as ϑ, and . The formula δ′ = ΔαS + ΔβT is generalized to Rn by means of the tensor product of distributions, and it follows from this formula that there is no discontinuous translation-invariant linear form on any of the spaces (Rn), (Rn), (Rn) or their duals. The same thing is also proved for (Tn) and its dual where Tn denotes the n-dimensional torus group.