Mahler's polynomial approximation for odd and even p-adic functions

Mahler's theorem says that, for every prime p  , the binomial polynomials form an orthonormal basis of the Banach space C(Zp,Qp)C(Zp,Qp) of continuous functions from ZpZp to QpQp. Recently, replacing QpQp by a local field K   and ZpZp by the valuation ring V of K, Klinger and Marshall constructed generalized binomial polynomials such that these odd (resp. even) binomial polynomials form an orthonormal basis of the space of odd (resp. even) continuous functions from V to K  . In this paper, we prove a similar result for odd and even functions in a more general framework by considering the Banach space C(E,K)C(E,K) of continuous functions, where K is any valued field and E is any symmetric regular compact subset of K.