On the Titchmarsh convolution theorem

Abstract

Let M be the set of all finite complex-valued Borel measures μ≢0 on ℝ. Set ℓ(μ) inf(supp μ). The classical Titchmarsh convolution theorem claims that if: (i) μj ∈ M, (ii) ℓ(μj) > - ∞, j = 1,. . . , n, then ℓ(μ1) + ⋯ + ℓ(μn) = ℓ(μ1 * ⋯ * μn). The condition (ii) cannot be omitted. In 80's, it had been shown that (ii) can be replaced with sufficiently rapid decay of the measures μj at - ∞ and the best possible condition of this form had been found. We show that the last condition can be weakened if we dealing with linearly dependent measures μj, and find the best possible condition in this case. © 2000 Académie des ciences/Éditions scientifiques et médicales Elsevier SAS.