E-sequences and the Stone-Weierstrass Theorem

For discrete valuation domain V with finite residue field, we define a variation of the V.W.D.W.O. sequence of Cahen and Chabert in order to construct V-bases for the algebra of even integer-valued polynomials on V and the module of odd integer-valued polynomials on V. Using these bases, we prove a version of the Stone-Weierstrass Theorem for V, namely, that every even (respectively odd) continuous function on the completion Vˆ can be approximated by means of even (respectively odd) integer-valued polynomials on V. Using these approximations, we give series expansions for all even (respectively odd) continuous functions on Vˆ, analogous to results of Mahler for the p-adic integers.