he calculus,
associated with piecewise polynomial manipulation, is covered in the section
(
Calculus behind the poly
module
). The results are implemented and tested in the Python "poly"
module. In this section we derive a formula for convolution of piecewise
polynomials in the case of the formula
(
Piecewise polynomial
representation
).
We start from the proposition
(
Convolution of polynomials 1
).
Let
where the notation
was introduced in the section (
Wavelet
analysis
) and
.
In the notation of the proposition
(
Convolution of polynomials
1
),
Then
Note
that
and we arrive
to
Let
then
We make the change of summation index
,
then
We change the order of summation, see the picture
(
Order of summation 5
).
|
Order of summation 5
|
We
have
We make the change
in the second
term.
We summarize our findings so far.
The integrals
may be evaluated using the proposition
(
Convolution of polynomials 2
).
Let
then
and we apply the proposition
(
Convolution of polynomials 2
)
with
,
,
,
.
We perform a similar calculation for
:
thus
We
continue
We summarize our recipe for calculation of convolution.
|
