o far we have
run the cascade procedure (
Cascade algorithm
)
according to the definition. On every step we produce an approximation
while we intend
for
in
where
is a scaling function of finite support. Thus, we roughly double the number of
pieces in a piecewise polynomial representation of
on every step.
Therefore, we would like to periodically interrupt the cascade procedure to
replace
with some
for some significant
so that we would still have
for small
.
Then we would restart the cascade procedure from
.
Previously, see section
(
Convergence of cascade
procedure
), we experimented to establish optimality of piecewise quadratic
representation of
.
Below we derive a recipe for the transformation
in such case.
Let
for some
.
Our intention is to use
to approximate some function
in
.
We drop the parameter
from notation in this
section:
We use quadratic
polynomials:
The cascade procedure preserves smoothness of the initial approximation.
Hence, we assume that we have continuity of first derivative, see the section
(
Convergence of cascade
procedure
), and intend to preserve such property. Furthermore, we regard
this situation in context of a large piecewise quadratic representation: we
have a lot of pieces
,
we single out these four and aim to transform them. Thus, the boundary values
at
and
come as input parameters. We arrive to the following
requirements.
where the
are input parameters. We have 12 parameters
and 10 conditions. At this point it should be apparent why we chose 4
intervals: this is the minimal even number of intervals that leaves out any
freedom. We introduce two parameters
to represent our degrees of
freedom:
Piecewise Quadratic
Polynomial picture

Next, we express the coefficients
as functions of the input data
and the control parameters
.
The following are the equations we are
solving:


(Quadratic polynomials 1)



(Quadratic polynomials 2)



(Quadratic polynomials 3)



(Quadratic polynomials 4)



(Quadratic polynomials 5)

We will be repeatedly solving the following partial
problem:
where we want to express
and
as functions of other variables. We multiply the second equation with
and add the two
equations:
Hence,


(Partial solution)

We apply the recipe (
Partial solution
) to
(
Quadratic polynomials 1
) and
(
Quadratic polynomials 3
) and
obtain
We substitute these into (
Quadratic
polynomials
2
):
or
We
simplify:
We introduce the notation
then
or
We add/subtract the equations and
obtain
or
Consequently,
We obtain resolution for the equations
(
Quadratic polynomials
3
)(
Quadratic polynomials 5
) by
making the
replacements
Thus
We summarize our findings.
