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Tauberian Theorems with Remainder
Author(s) -
Omey E.
Publication year - 1985
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-32.1.116
Subject(s) - remainder , corollary , mathematics , laplace transform , combinatorics , discrete mathematics , arithmetic , mathematical analysis
Suppose that f , g : R + → R + are non‐decreasing functions of x with finite Laplace transformsf ^ ( s )andg ^ ( s ) , respectively. In this paper we discuss conditions under which, as x → ∞,f ^ ( 1 / x ) − f ^ ( 1 / x ) = 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 → ∞, that f(tx)/f(t) = x rβ + O(a(x)) if and only iff ^ ( 1 / t x ) / f ^ ( 1 / t x ) = x rβ + O(a(x)) .