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Journal of the London Mathematical Society 1985 s2-32(1):116-132; doi:10.1112/jlms/s2-32.1.116
© 1985 by London Mathematical Society
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© Oxford University Press

Tauberian Theorems with Remainder

E. Omey

Economische Hogeschool Sint-Aloysius Broekstraat 113, 1000 Brussel, Belgium

Suppose that f, g: R+ -> R+ are non-decreasing functions of x with finite Laplace transforms Formula and Formula, respectively.

In this paper we discuss conditions under which, as x -> {infty}, Formula–Formula = O(a(x)) implies that f(x)–g(x) = O(b(x)) for certain classes of functions g(x), a(x) and b(x), thereby extending a result of Ingham.

As a corollary we also obtain an Abel-Tauber theorem for regularly varying functions with remainder, that is, under certain conditions on f(x) and a(x) we prove, as x -> {infty}, that f(tx)/f(t) = xrß + O(a(x)) if and only if Formula/Formula = xrß + O(a(x)).


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