2
The basic lower bound conditions that we propose are as
follows.
i. There exists c op i, d op’ j, such that for all x
∈
dom(f), c|x|
d
op’’ f(x). Here op,op’
∈
{<,>,
≤
,
≥
,=}, op’’
∈
{<,
≤
}, i,j
∈
{0,1/2,1,3/2,2}, and | | is the l
∞
norm, the l
1
norm, or the l
2
norm.
The basic upper bound conditions that we propose are as
follows.
ii. There exists c op i, d op’ j, such that for all x
∈
dom(f), c|x|
d
op’’ f(x). Here op,op’
∈
{<,>,
≤
,
≥
,=}, op’’
∈
{>,
≥
}, i,j
∈
{0,1/2,1,3/2,2}, and | | is the l
∞
norm, the l
1
norm, or the l
2
norm.
Each of these conditions in i,ii above results from the
choice of 6 parameters. Note that some of the choices of
parameters result in degenerate conditions.
Each of these basic lower bound conditions and basic upper
bound conditions can be modified by using “for all but
finitely many x” instead of “for all x”. This doubles the
number of lower and upper conditions, and the resulting
conditions are called the lower bound conditions and the
upper bound conditions.
The bounding conditions consist of a conjunction of zero or
more conditions, each of which is either a basic lower
bound condition or a basic upper bound condition. Here any
of these basic lower bound and upper bound conditions an be
modified as in the previous paragraph.