A note on generalized Tauberian theorems
1951; American Mathematical Society; Volume: 2; Issue: 3 Linguagem: Inglês
10.1090/s0002-9939-1951-0041954-4
ISSN1088-6826
Autores Tópico(s)Mathematical and Theoretical Analysis
ResumoIntroduction.Suppose that A(u) is a function of bounded variation in every finite interval of w^O, A(G) =0,' and that the 4>-transform of A(u), namely, $0) = f 4>(ut)d{A(u)}, Jo is convergent for t>0, the function <b(u) satisfying the following conditions.C(i) For m^O, <p(u) is positive, continuous, and monotonie decreasing;(ii) (0) = l, fx((l>(u)/u)du is convergent;(iii) for m^O, (u) has a continuous derivative -yp(u) which is, on account of (i), negative and such that *(«) = f *(x)dx;2 (iv) for m^O, \p(u) is monotonie decreasing and has a continuous derivative.Then a generalized Tauberian theorem in the sense in which the expression is used in the present context is a "converse" theorem which enables us to relate the behaviour of A(u) as u-»oo to that of 3>(£) as /->+0, under a suitable Tauberian condition.In the usual terminology, such a theorem may be called an "O-inversionssatz" when it has a Tauberian ol-or o/e-hypothesis and a conclusion which relates lim sup (or inf) A(u), u-»oo, to lim sup (or inf) $>(/), /-»+o; an "o-inversionssatz," when it assumes a Tauberian Ol-or O/e-condition and deduces the convergence of A(u) as u-»oo from that of <£(<) as /-»+0.It is the object of this note to show that the classical inversion theorems of Karamata and Ramaswami, of the two kinds mentioned
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