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The digit principle. (English) Zbl 1017.11061

Several constructions in function field arithmetic involve extensions from linear objects using digit expansions, for instance, Carlitz polynomials. In this paper, the technique of using digit expansions is described as a method of constructing orthonormal bases in spaces of continuous functions.
Let \(K\) be a local field of nonzero characteristic, with ring of integers \(\mathcal O\) and residue field \(F\). We consider the \(K\)-Banach space \(C ({\mathcal O}, K)\) of continuous functions from \(\mathcal O\) to \(K\) with the sup-norm and \(\hom _F ({\mathcal O},K)\) the closed subspace of continuous \(F\)-linear functions from \(\mathcal O\) to \(K\). A sequence is an orthonormal basis of \(\hom _F ({\mathcal O},K)\) if and only if the sequence of reductions is an algebraic basis of \(\hom _F ({\mathcal O},F)\). Given a sequence \(\{e_i\}\) which is an orthonormal basis of \(\hom _F ({\mathcal O},K)\), we define the extension of the \(e_j\) by \(q\)-digit expansion (where \(q = |F|\)) as the sequence \(\{f_i\}\) by writing \(i\) in base \(q\), \(i=c_0+c_1q+\cdots+c _{n-1} q^{n-1}\), \(0\leq c_j\leq q-1\) and \[ f _i := e _0^{c _0}e _1^{c _1} \cdots e _{n - 1}^{c _{n - 1}}. \] The author proves first the digit principle in positive characteristic: the extension of an orthonormal basis of \(\hom_F({\mathcal O},K)\) via \(q\)-digit expansions provides an orthonormal basis for \(C({\mathcal O},K)\). The digit principle does not apply in characteristic \(0\) as formulated. However, using a remark of A. Baker [J. Lond. Math. Soc. (2) 33, 414-420 (1986; Zbl 0562.12016)] the author shows that replacing the linear conditions with a property that comes up in the proof in the positive characteristic case, the digit principle extends to characteristic \(0\).
The digit principle is applied to several situations such as: Carlitz functions are an orthonormal basis of \(C (\mathbb{F} _q [[T]], \mathbb{F}_q ((T)))\); the digit expansions of hyperdifferential functions form an orthonormal basis of \(C (\mathbb{F}_q [[T]], \mathbb{F}_q ((T)))\) (this is the original motivation of the paper); orthonormal basis of \(C(\mathbb{Z}_p, \mathbb{Q}_p)\); orthonormal bases related to Lubin-Tate groups, etc. Finally a concrete model is obtained for the continuous functions on the integers of a local field as a quotient of a Tate algebra in countably many variables.

MSC:

11S85 Other nonanalytic theory
46E15 Banach spaces of continuous, differentiable or analytic functions
11T99 Finite fields and commutative rings (number-theoretic aspects)
11R99 Algebraic number theory: global fields

Citations:

Zbl 0562.12016
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References:

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