I. Python Object Browser.
 II. Python to R Communicator.
 III. Manipulation of piecewise polynomial functions.
 1 History of changes (poly).
 2 Installation of the poly module.
 3 Introduction into the poly module.
 4 Calculus behind the poly module.
 IV. Building C++ projects.

## Calculus behind the poly module.

o understand the code of the functions S(),T(), review the formulas ( Property of scale and transport 1 )-( Property of scale and transport 7 ).

The following propositions are reflected in the classes Poly and PiecewisePoly.

Proposition

(Transport of polynomial) Suppose the polynomial is given by the sequence : then is given by ,

Proof

Recall that According to the definition ( Scale and transport operators 2 ),

Proposition

(Scale of polynomial) Suppose the polynomial is given by the sequence : then is given by ,

Proof

According to the definition ( Scale and transport operators 2 ),

Proposition

(Multiplication of polynomials) Suppose the polynomials and are given by the sequences and then the product is given by the sequence where

Proof

We calculate Make the change , : We now change the order of summation, see the picture ( Order of summation 0 ).
Order of summation 0

Proposition

(Integration of polynomials) Suppose the polynomial is given by the sequence : then

Proposition

(Convolution of polynomials 1) Let for some polynomials and intervals . Then where and we use the convention

Proof

We calculate There are several mutually exclusive cases, see the picture ( Piecewise convolution ).

Piecewise convolution.

Let then the integral might have a non empty interval of integration in the following mutually exclusive cases: We transform the above inequalities: Therefore the integral takes the form

Proposition

(Convolution of polynomials 2) Let then

Proof

We calculate We change the order of summation, see the figure ( Order of summation 1 ).

Order of summation 1

We introduce the notation then We permute the order of and -summations, see the figure ( Order of summation 2 ).

Order of summation 2

We permute the order of and -summations, see the figure ( Order of summation 3 ).

Order of summation 3

We make the change , . We permute the order of and -summations, see the figure ( Order of summation 4 ).

Order of summation 4