Continuous \(\pi\)-adic functions and \(\pi\)-derivations
Let \(K\) be a local field and \({\mathcal O}\) its ring of integers. Let \(\pi\) be a uniformizer of \({\mathcal O}\) and let \(\phi\) be an automorphism of \(K\) lifting the \(q\)-power map on the residue field \({\mathcal O}/\pi\). Let \(\delta\) be the \(\pi\)-derivation \(\delta(a) =(\phi(a)- a^q)/\pi\). The aim of this paper is to prove that the set \(\{\delta _{\underline{i}}\}\), where \({\underline{i}} = (i_0, i_1, \ldots)\) is a multi-index with integer coordinates satisfying \(0\leq i_j\leq |{\mathcal O}/(\pi)|-1\) and \(i_j =0\) for all \(j\) large enough, is an orthonormal basis of the continuous map space \(C({\mathcal O}, {\mathcal O})\). This result is analogous to a result of \textit{K. Mahler} [Introduction to \(p\)-adic numbers and their functions. Cambridge University Press (1973; Zbl 0249.12015), IX, 91] which states that the binomial coefficient functions \(\binom{a}{n}\) form an orthonormal basis of \(C({\mathbb Z}_p,{\mathbb Z}_p)\). After this paper was submitted, the author received a preprint of a paper by \textit{K. Conrad} [J. Number Theory 84, 230-257 (2000; Zbl 1017.11045)] where a general theory of orthonormal basis based on the ``the digit principle'' has been developed. In particular the theorem of the paper under review follows from the results of Conrad.
scientific article; zbMATH DE number 1533154