The basic purpose of the analysis is to derive an n-dimensional analogue of the Euler Maclaurin expansion valid for any quadrature role Q for the simplex.We choose for our calculation the simplex $S_n :x_i > 0;\Sigma _{i = 1}^n x_i < 1$ which lies within the hypercube $H_n :0 < x_i < 1$. In this paper we construct a piecewise polynomial approximation $P({\bf x})$ to a function $G({\bf x})$ which coincides with a given analytic function $\phi ({\bf x})$ within $S_n$ but which is zero in the part of $H_n$ not contained in $S_n$. This representation has the form $\Sigma _j \phi _j \beta _j ({\bf x})$ where $\phi_j$ are constants depending on $\phi ({\bf x})$ alone and $\beta _j ({bf x})$ are independent of $\phi ({\bf x})$, being the product of n periodic Bernoulli functions $\bar B_q (t)$. The arguments t are linear expressions in $x_i$, these being arranged so that discontinuities in $\bar B_q (t)$ give rise to the required discontinuity in $G ({\bf x})$.This representation has specified polynomial propert...