1. Translation-invariant linear forms and a formula for the dirac measure
- Author
-
Gary H. Meisters
- Subjects
Discrete mathematics ,39A05 ,Pure mathematics ,Dirac measure ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,28A30 ,Space (mathematics) ,symbols.namesake ,Tensor product ,Linear form ,10F25 ,symbols ,Order (group theory) ,46H10 ,Dual polyhedron ,Invariant measure ,Algebraic number ,46F10 ,42A68 ,Analysis ,Real number ,Mathematics - Abstract
It is shown in this paper (Theorem 1) that if α and β are real numbers such that α gb 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 α gb 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 D, E, S or their duals then ϑ ∗ S and ϑ ∗ T belong to the same space as ϑ, and ϑ′ = Δ α (ϑ ∗ S) + Δ β (ϑ ∗ T) . 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 D (Rn), E (Rn), S (Rn) or their duals. The same thing is also proved for E (Tn) and its dual where Tn denotes the n-dimensional torus group.
- Published
- 1971
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