(2012-08-01) Fundamentals of Asymptotics
Only zero is asymptotic to zero.
Let's first consider numerical functions
(where division makes sense):
Numerical Functions :
Two numerical functions f and g
are said to be asymptotic (or equivalent)
to each other in the neighborhood of some limit point L when the
ratio f (x) / g (x) tends to 1
as x tends to L. In other words, the following two notations
are equivalent, by definition:
f (x) ~ g (x)
( x ® L )
f (x) / g (x)
Likewise, the statement "
f (x) is negligible  compared to
g (x) as x tends to L "
is denoted or defined as follows:
f (x) << g (x)
( x ® L )
f (x) / g (x)
Note that this relation is unrelated to ordering in the real line.
For example, both of the following relations are true as x tends to zero:
-1 < x2
x2 << -1
You can manipulate algebraically an asymptotic equivalence exactly as you would
an ordinary equation, except that you're not allowed
to transpose everything to one side of the equation!
Nothing (but zero itself) is asymptotic to zero...
Extension to Vectorial Functions :
For vectorial functions, the symmetry in the above definitions must be broken.
Negligibility is not difficult to define in a normed vector space:
One quantity is negligible compared to another when the norm of the first
is negligible compared to the norm of the other.
With this in mind, we can promote to a definition among vectors what's
a simple characteristic theorem for equivalent scalars quantities
(with the definitions given above):
Definitions for Vectorial Asymptotics
As x tends to L, one vectorial quantity
f (x) is said to be negligible compared to
another quantity g (x) when the ratio
|| f (x) || / || g (x) ||
has a limit of zero.
Two quantities are asymptotically equivalent to each other
("asymptotic to" or "equivalent to", for short)
if their difference is negligible compared to their sum.
Is zero asymptotic to zero?
In asymptotics, "zero" is any function which is identically equal to 0
(the null vector) in some
neighborhood of the relevant limit point.
The following relations are valid whenever f is a nonzero quantity:
0 << f
f ~ f
By convention, we retain the validity of those two for zero quantities:
Only zero is negligible compared to zero. Only zero is equivalent to zero.
(2012-10-04) Asymptotic expansions about a limit point.
Asymptotic expansions may or may not be convergent.
Stieltjes Functions & Moments
Well before the more general notion of distributions was devised
(in 1944, by my late teacher Laurent Schwartz)
the Dutch mathematician
Thomas Stieltjes considered measures as generalized
derivatives of functions of bounded variations of a real variable.
Such functions are differences of two monotonous bounded functions;
they need not be differentiable or continuous.
(Stieltjes got his doctorate in Paris,
under Hermite and Darboux.)
Let's define a weight function r
as a nonnegative function of a nonnegative
variable which has a moment of order n,
expressed by the following convergent integral,
for any nonnegative integer n :
r(t) t n dt
To any such weight function is associated a Stieltjes function defined by: