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.