Master Dissertation

Chapter 6
Regularly Varying Functions
Before going on in the splitting process we need to consider some
properties of a class of functions which will become important.
Definition 6.1. A positive function ρ(t) is called regularly varying at
t = 0, if it is measurable in (0, t0] for some t0 > 0, and there exists ω ∈ R
such that the limit
ρ(at)
lim = aω
t→0 ρ(t)
exists for all a > 0. In this case ω is called the order of ρ. If ω = 0 the
function is said to be slowly varying.
(6.1)
Note that a regularly varying function always can be reduced to be slowly
varying. That is, let
ρ(t) = t ω r(t). (6.2)
Then (6.1) gives
r(at)
lim = 1, (6.3)
t→0 r(t)
for all a > 0.
The following representation theorem will be important for our use.
Theorem 6.2. Let r : (0, t0] → R+ be a slowly varying function. Then
there exists some t1 ∈ (0, t0) such that
t1 h(s)
r(t) = exp η(t) +
t s ds
holds for all 0 < t < t1. Here η is bounded and measurable on (0, t1], and
η → c for t → 0 (|c| < ∞). Further h(t) is continuous on (0, t1] and
h(t) → 0 for t → 0.
Before proving the theorem we will need some lemmas.
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