Abstract
Mahler's theorem says that, for every prime p, the binomial polynomials form an orthonormal basis of the Banach space C(Zp,Qp) of continuous functions from Zp to Qp. Recently, replacing Qp by a local field K and Zp 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) of continuous functions, where K is any valued field and E is any symmetric regular compact subset of K.
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