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Around Widder’s characterization of the Laplace transform of an element of L ( + )

Jan Kisyński — 2000

Annales Polonici Mathematici

Let ϰ be a positive, continuous, submultiplicative function on + such that l i m t e - ω t t - α ϰ ( t ) = a for some ω ∈ ℝ, α ∈ + ¯ and a + . For every λ ∈ (ω,∞) let ϕ λ ( t ) = e - λ t for t + . Let L ϰ 1 ( + ) be the space of functions Lebesgue integrable on + with weight ϰ , and let E be a Banach space. Consider the map ϕ : ( ω , ) λ ϕ λ L ϰ 1 ( + ) . Theorem 5.1 of the present paper characterizes the range of the linear map T T ϕ defined on L ( L ϰ 1 ( + ) ; E ) , generalizing a result established by B. Hennig and F. Neubrander for ϰ ( t ) = e ω t . If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...

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