he crudification operator was introduced in the section
(
Crudification
of piecewise-quadratic representation
). We need to adapt the result of
that section to the changes in representation. Most of the definition
(
Crudification operator
) stays the same
except for reference to the proposition
(
Quadratic piecewise
interpolation
) that needs to be replaced with an updated proposition. We
perform the necessary calculations and present an updated summary.

Proposition

(Quadratic piecewise
interpolation 2) A piecewise polynomial
of the
form
satisfies the
conditions
for given
if and only
if

Proof

The proposition (
Quadratic
piecewise interpolation
) provides expressions for the polynomials
such
that
We convert the result to the localized
representation:
thus
where the expressions for
are known from the proposition
(
Quadratic piecewise
interpolation
). The rest of the calculation is performed by the
Mathematica script located in
OTSProjects/Cuda/PiecewisePoly/pyd/src/crudification.nb.

Definition

(Crudification operator 2) Let
be the class of piecewise quadratic functions with finite support defined on
:
We define a transformation
according to the
rule
where the piecewise polynomial
is
constructed according to the following procedure.