On $P$-orderings, rings of integer-valued polynomials, and ultrametric analysis

We introduce two new notions of ``$P$-ordering'' and use them to define a three-parameter generalization of the usual factorial function. We then apply these notions of $P$-orderings and factorials to some classical problems in two distinct areas, namely: 1) the study of integer-valued polynomials and 2) $P$-adic analysis.

Specifically, we first use these notions of $P$-orderings and factorials to construct explicit Pólya-style regular bases for two natural families of rings of integer-valued polynomials defined on an arbitrary subset of a Dedekind domain.

Second, we classify ``smooth'' functions on an arbitrary compact subset $S$ of a local field, by constructing explicit interpolation series (i.e., orthonormal bases) for the Banach space of functions on $S$ satisfying any desired conditions of continuous differentiability or local analyticity. Our constructions thus extend Mahler's Theorem (classifying the functions that are continuous on $\mathbb{Z}_p$) to a very general setting. In particular, our constructions prove that, for any $\epsilon>0$, the functions in any of the above Banach spaces can be $\epsilon$-approximated by polynomials (with respect to their respective Banach norms). Thus we obtain the non-Archimedean analogues of the classical polynomial approximation theorems in real and complex analysis proven by Weierstrass, de la Vallée-Poussin, and Bernstein. Our proofs are effective.