For
even we have
,
for some integer
and an odd integer
.
We use the proposition (
Scaling equation
)
times:
thus
and we repeat the calculation of the odd
case:

Remark

Note
that
is a 1-periodic function. Hence, we may consider the Fourier coefficients of
:
We make a change
.
Under conditions of the last proposition,
.
Thus

1.
is compactly supported and (therefore)
is finite,

2.
,
.

Then there exist polynomials
of degree
such
that
for
.

Proof

We introduce the convenience
notation
then the proposition (
Reproduction
of polynomials 2
)
provides
For
we have
For
we
have
so that
For
we
have
We use results of previous
steps.
We continue similarly for all
up to
.