We introduce two new notions of “
P
P
-ordering” and use them to define a three-parameter generalization of the usual factorial function. We then apply these notions of
P
P
-orderings and factorials to some classical problems in two distinct areas, namely: 1) the study of integer-valued polynomials and 2)
P
P
-adic analysis.
Specifically, we first use these notions of
P
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
S
of a local field, by constructing explicit interpolation series (i.e., orthonormal bases) for the Banach space of functions on
S
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
Z
p
\mathbb {Z}_p
) to a very general setting. In particular, our constructions prove that, for any
ϵ
>
0
\epsilon >0
, the functions in any of the above Banach spaces can be
ϵ
\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.