Adelic versions of the Weierstrass approximation theorem
Section snippets
Introduction: the classical p-adic versions of Weierstrass theorem
A well known p-adic version of the Weierstrass polynomial approximation theorem due to Dieudonné [9] states the following:
Theorem 1.1 For every compact subset E of , the ring of polynomial functions is dense in the ring of continuous functions from E into with respect to the uniform convergence topology.
If we restrict to functions with values in , we have to consider the subring of polynomial functions whose values on E are in , namely the ring , and then,
The adelic framework
Let denote the topological ring of finite adèles of (see for example [10, Chapter 1]). Recall that, as a subset, is the restricted product of the fields with respect to the rings , as p ranges through the set of primes , that is: The topology on is the restricted product topology characterized by the fact that the following subsets form a basis of open subsets: where is an open subset of and
First adelic Weierstrass theorems
In the previous section we remarked that for our considerations without loss of generality we may restrict our proofs to compact subsets E of (Lemma 2.2).
Definition 3.1 Let E be a subset of . The subring of formed by the rational polynomial functions which are integer valued on the subset E, that is, , is denoted by . More precisely:
This notation is a particular case of the following that we will use in the
Bases of the -module
In order to obtain bases analogous to those of Theorem 1.3, Theorem 1.4, if there are some, for the -Banach space , we are looking now for bases of the -module .
For every compact subset E of , we have that is a -module. Does this -module admit a basis? As noticed in Remark 3.2, we may assume for simplicity that the compact subset E is of the form , and hence, that (see also [8, (6.1)]): where
Polynomials with adelic coefficients
Let us consider now polynomials with coefficients in : As before, let be the canonical projection from to . For a polynomial in we consider its components , so that we have: corresponding to the containment . Note that the last containment is strict, even if we consider the restricted product of the 's with respect to
Extension of Bhargava–Kedlaya's theorem
Recall that, in , a sequence converges to if and only if, for each , the sequence converges to in . In the same way, in , a sequence converges uniformly to if and only if, for each , converges uniformly to in .
It follows from the previous section that Mahler's result (Theorem 1.3) extends in the following way:
Proposition 6.1 Every may be uniquely written as a
Acknowledgement
The authors wish to thank the referee for carefully reading the paper. The authors have been supported by grant “Bando Giovani Studiosi 2013”, Project title “Integer-valued polynomials over algebras” Prot. GRIC13X60S of the University of Padova.
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