I. Introduction into GPU programming.
 II. Exception safe dynamic memory handling in Cuda project.
 III. Calculation of partial sums in parallel.
 IV. Manipulation of piecewise polynomial functions in parallel.
 1 History of changes (PiecewisePoly).
 2 Calculus behind the PiecewisePoly project.
 3 Code structure (PiecewisePoly project).
 4 Python scripting for PiecewisePoly project.
 V. Manipulation of localized piecewise polynomial functions in parallel.

## Calculus behind the PiecewisePoly project.

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.

Proposition

(Convolution of polynomials 3) Let then where

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.

Proposition

(Convolution of polynomials 4) Let for , , .

Then where