Normal Hilbert functions of one-dimensional local rings

Let (R, m) be a one-dimensional reduced (Noetherian) local ring with finite integral closure ((R) over bar, M-1,..., M-t). We assume further that R/m similar or equal to (R) over bar /M-i for each i and that Card(R/m) greater than or equal to t. We study for such a ring R the associated graded ring and the Hilbert series, with respect to the normal filtration of an m-primary ideal I, R superset of or equal to (I) over bar superset of or equal to (I-2) over bar superset of or equal to (...). We make use of the value semigroup of R and in particular of some results of (7)