sentences:  Now consider the finite-dimensional space of discontinuous piecewise polynomial functions over the spatial domain formula_2 restricted to a discrete triangulation formula_7, written as
  
  The index formula_13 should show the relation to an underlying discretization given by formula_14.Note here that formula_9 is not uniquely defined at the intersection points formula_16.At first we make use of a specific polynomial basis on the interval formula_17, the Legendre polynomials formula_18, i.e.,
  Note especially the orthogonality relations 
  Transformation onto the interval formula_21, and normalization is achieved by functions formula_22
  which fulfill the orthonormality relation
  Transformation onto an interval formula_10 is given by formula_26
  which fulfill
  For formula_29-normalization we define formula_30, and for formula_31-normalization we define formula_32, s.t.