Published in

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

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Translation-invariant linear forms and a formula for the Dirac measure

Journal article published in 1971 by Gary H. Meisters
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

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.