Preliminaries

Abstract

In this chapter we will introduce some of the notations and the technical setup of our book. The rth symmetric difference of a function f is given by

$$ \begin{gathered} \Delta _h^1 f(x) = \Delta _h f(x) = f\left( {x + \frac{h} {2}} \right) - f\left( {x - \frac{h} {2}} \right), \hfill \\ \Delta _h^r f(x) = \Delta _h \left( {\Delta _h^{r - 1} f\left( x \right)} \right), \hfill \\ \end{gathered} $$

(1.1.1)

or equivalently by

$$\Delta _h^rf(x) = \sum\limits_{k = 0}^r {{{( - 1)}^k}(_k^r)f(x + r(h/2) - kh).} $$

(1.1.2)