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A Condition, Necessary and Sufficient, for the Representability as a Laplace Transform of a Distribution

  • Gustav Doetsch

Abstract

It is a characteristic of the L-transform of a function that it tends towards zero when the variable s, two-dimensionally in an angular region | arcs | ≦ ψ < π/2, tends towards ∞ (compare Theorem 23.2, Addendum). This property is true in a whole half-plane ℜs ≧ xo, provided L{f} converges absolutely in ℜsx 0 (compare Theorem 23.7). The examples L{δ(n)} = s n (n = 0, 1, 2, . . . ), ℜs > 0, demonstrate that the ℒ-transforms of distributions need not possess this property. Actually, these transforms tend towards ∞ when s → ∞ in ℜs > 0; however, not more strongly than a power of s. We shall show that the L-transforms of distributions are completely characterized by the property of being majorized by powers of s. The following Theorems 29.1 and 29.2 will substantiate this claim. Concepts and terminology involved in this process are explained in Chapter 12.

Keywords

Stochastic Process Analytic Function Probability Theory Laplace Transform Circular Disc 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Gustav Doetsch
    • 1
  1. 1.Emeritus of MathematicsUniversity of FreiburgFreiburg i. Br.Germany

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