Conrad, Keith The digit principle. (English) Zbl 1017.11061 J. Number Theory 84, No. 2, 230-257 (2000). 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. Reviewer: Gabriel D.Villa-Salvador (Mexico, D.F.) Cited in 2 ReviewsCited in 31 Documents 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 Keywords:digit principle; local field; orthonormal basis; Carlitz polynomial; hyperdifferential operator; Lubin-Tate group; Tate algebra; spaces of continuous functions; Banach space; \(q\)-digit expansion; positive characteristic Citations:Zbl 0562.12016 PDFBibTeX XMLCite \textit{K. Conrad}, J. Number Theory 84, No. 2, 230--257 (2000; Zbl 1017.11061) Full Text: DOI arXiv References: [1] Amice, Y., Interpolation \(p\)-adique, Bull. Soc. Math. Fr., 92, 117-180 (1964) · Zbl 0158.30201 [2] Baker, A., \(p\)-Adic continuous functions on rings of integers and a theorem of K. Mahler, J. London Math. Soc., 33, 414-420 (1986) · Zbl 0562.12016 [3] Bosch, S.; Güntzer, U.; Remmert, R., Non-Archimedean Analysis (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0539.14017 [4] Cahen, P.-J; Chabert, J.-L, Integer-Valued Polynomials (1997), Amer. Math. Society: Amer. Math. Society Providence [5] Car, M., Pólya’s theorem for \(F}_q[T]\), J. Number Theory, 66, 148-171 (1997) · Zbl 0886.11034 [6] De Smedt, S., Some new bases for \(p\)-adic continuous functions, Indag. Math. (N.S.), 4, 91-98 (1993) · Zbl 0791.46061 [7] Goss, D., Fourier series, measures, and divided power series in the theory of function fields, K-Theory, 1, 533-555 (1989) · Zbl 0675.12006 [8] Goss, D., Basic Structures of Function Field Arithmetic (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0874.11004 [9] Hasse, H.; Schmidt, F. K., Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten, J. Reine Angew. Math., 177, 215-237 (1937) · Zbl 0017.10101 [10] Jeong, S., Diophantine Problems in Function Fields of Positive Characteristic (May 1999), Univ. Texas: Univ. Texas Austin [11] S. Jeong, On orthonormal bases of continuous functions on power series rings, preprint.; S. Jeong, On orthonormal bases of continuous functions on power series rings, preprint. [12] Kawahara, Y.; Yokoyama, Y., On higher differentials in commutative rings, TRU Math., 2, 12-30 (1966) · Zbl 0158.04302 [13] Lang, S., Cyclotomic Fields I and II (1990), Springer-Verlag: Springer-Verlag New York [14] Okugawa, K., Differential Algebra of Nonzero Characteristic (1987), Kinokuniya: Kinokuniya Tokyo · Zbl 0757.12004 [15] Serre, J.-P, Endomorphismes complètement continus des espaces de Banach \(p\)-adiques, Publ. Math. IHES, 12, 69-85 (1962) · Zbl 0104.33601 [16] Serre, J.-P, A Course in Arithmetic (1973), Springer-Verlag: Springer-Verlag New York · Zbl 0256.12001 [17] Snyder, B., Hyperdifferential Operators on Function Fields and Their Applications (1999), Ohio State Univ [18] Tateyama, K., Continuous functions on discrete valuation rings, J. Number Theory, 75, 23-33 (1999) · Zbl 0938.12003 [19] Teichmüller, O., Differentialrechnung bei Charakteristik \(p\), J. Reine Angew. Math., 175, 89-99 (1936) · JFM 62.0114.01 [20] van Rooij, A. C.M, Non-Archimedean Functional Analysis (1978), Marcel Dekker: Marcel Dekker New York · Zbl 0396.46061 [21] Voloch, J. F., Differential operators and interpolation series in power series fields, J. Number Theory, 71, 106-108 (1998) · Zbl 0999.12013 [22] Wagner, C., Interpolation theorems for continuous functions on \(π\)-adic completions of \(GF (q, x)\), Acta Arith., 17, 389-406 (1971) · Zbl 0223.12009 [23] Yang, Z., Locally analytic functions over completions of \(F}_r[U]\), J. Number Theory, 73, 451-458 (1998) · Zbl 1029.11065 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.