Adelic versions of the Weierstrass approximation theorem

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Abstract

Let E_=pPEp be a compact subset of Zˆ=pPZp and denote by C(E_,Zˆ) the ring of continuous functions from E_ into Zˆ. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring IntQ(E_,Zˆ):={f(x)Q[x]|f(E_)Zˆ} is dense in the product pPC(Ep,Zp) for the uniform convergence topology. We also obtain an analogous statement for general compact subsets of Zˆ.

Secondly, under the hypothesis that, for each n0, #(Ep(modp))>n for all but finitely many primes p, we prove the existence of regular bases of the Z-module IntQ(E_,Zˆ), and show that, for such a basis {fn}n0, every function φ_ in pPC(Ep,Zp) may be uniquely written as a series n0c_nfn where c_nZˆ and limnc_n0. Moreover, we characterize the compact subsets E_ for which the ring IntQ(E_,Zˆ) admits a regular basis in Pólya's sense by means of an adelic notion of ordering which generalizes Bhargava's p-ordering.

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 Qp, the ring of polynomial functions Qp[x] is dense in the ring C(E,Qp) of continuous functions from E into Qp with respect to the uniform convergence topology.

If we restrict to functions with values in Zp, we have to consider the subring of polynomial functions whose values on E are in Zp, namely the ring Int(E,Zp)={fQp[x]|f(E)Zp}, and then,

The adelic framework

Let Af(Q) denote the topological ring of finite adèles of Q (see for example [10, Chapter 1]). Recall that, as a subset, Af(Q) is the restricted product of the fields Qp with respect to the rings Zp, as p ranges through the set of primes P, that is:Af(Q)={x_=(xp)pPQp|xpZp for all but finitely many pP}. The topology on Af(Q) is the restricted product topology characterized by the fact that the following subsets form a basis of open subsets: pPOp where Op is an open subset of Qp and Op=Zp

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 Zˆ (Lemma 2.2).

Definition 3.1

Let E be a subset of Zˆ. The subring of C(E,Zˆ) formed by the rational polynomial functions which are integer valued on the subset E, that is, C(E,Zˆ)Q[x], is denoted by IntQ(E,Zˆ)={fQ[x]|f(E)Zˆ}. More precisely:IntQ(E,Zˆ):={fQ[x]|α_=(αp)pE,pP,f(αp)Zp}.

This notation is a particular case of the following that we will use in the

Bases of the Z-module IntQ(E,Zˆ)

In order to obtain bases analogous to those of Theorem 1.3, Theorem 1.4, if there are some, for the Q-Banach space C(E,Af(Q))ρE(pPC(Ep,Qp)), we are looking now for bases of the Z-module IntQ(E,Zˆ).

For every compact subset E of Zˆ, we have that IntQ(E,Zˆ) is a Z-module. Does this Z-module admit a basis? As noticed in Remark 3.2, we may assume for simplicity that the compact subset E is of the form E_=pEp, and hence, that (see also [8, (6.1)]):IntQ(E_,Zˆ)=pPIntQ(Ep,Zp), where IntQ(Ep,Zp)={f

Polynomials with adelic coefficients

Let us consider now polynomials g_(x) with coefficients in Af(Q):g_(x)=k=0nγ_kxk, where γ_k=(γk,p)pPAf(Q). As before, let πp be the canonical projection from Af(Q)[x] to Qp[x]. For a polynomial g_(x) in Af(Q)[x] we consider its components πp(g_(x))=gp(x)Qp[x], so that we have:g_=(gp)pP, where gp(x)=k=0nγk,pxk with γk,pQp corresponding to the containment Af(Q)[x]pPQp[x]. Note that the last containment is strict, even if we consider the restricted product of the Qp[x]'s with respect to

Extension of Bhargava–Kedlaya's theorem

Recall that, in Zˆ, a sequence {α_n}n0={(αn,p)pP}n0 converges to α_=(αp)pP if and only if, for each pP, the sequence {αn,p}n0 converges to αp in Zp. In the same way, in C(E_,Zˆ), a sequence {φ_n}n0={(φn,p)pP}n0pPC(Ep,Zp) converges uniformly to φ_=(φp)pP if and only if, for each pP, {φn,p}n0 converges uniformly to φp in C(Ep,Zp).

It follows from the previous section that Mahler's result (Theorem 1.3) extends in the following way:

Proposition 6.1

Every φ_pPC(Zp,Zp) 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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