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