10
I. HECKE L-FUNCTIONS
An example of a distribution is given by a locally integrable function, e.g. if cp
is integrable, the linear functional T^ defined by
( 1 ^ , $ ) = / ip{x)$(x)dx
Jx
is continuous on C%°(X). When this is the case we identify the distribution X^
with the locally integrable function (p. Observe that the same distribution may be
associated with different functions, all of which agree outside a set of measure 0.
As the following example shows, not all distributions arise from locally integrable
functions.
DEFINITION 11. The Dirac distribution S is defined by the linear functional
(*,$) = $(0).
DEFINITION 12. A sequence of distributions {T
n
}
n G N
in V'{X) is said to
converge to a distribution T if (Tn, p) (T, ip) for any (p G C%°(X).
We say that a distribution T vanishes on an open set U when (T, $) = 0 for
all $ G C£°(X) with suppQ C U. It is clear that if a distribution T vanishes in a
neighborhood of every point of X, then T vanishes identically, i.e. (T, 3) = 0 for
all £ G C£°(X). A point x G X is called an essential point of the distribution T if
T 7^ 0 in every neighborhood of x.
DEFINITION 13. The support of a distribution T is the set of all its essential
points; it is denoted by suppT.
Since the support of a distribution is the complement of the largest open set in
which T = 0, suppT is a closed set.
EXAMPLE. The support of the Dirac distribution 5 is {0}.
A distribution T is said to be of order fc, if for every compact set K d , there
exists a constant C such that
|T($)| C J2 aupld"^, $ G C?(X),
\oc\k
where
da
is the differential operator corresponding to the multi-index a= (ai, c*2, •••)
and \a\ = a\ + «2 + The derivative of a distribution T in V{X) is defined by
the formula
(T
( 1 )
,$) = (T,0$).
More generally, for a multi-index a = (c*i, «2,. •) we put
(T
a
,$) =
(T,9a$).
If g is an arbitrary point in X, we denote by 5g the Dirac distribution concentrated
at g. The following result will be useful later on.
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