Orthonormal q-bases associated to some continuous linear operators on (ℤ _p ,K)

Abstract

Let K be a complete valued field, extension of the p-adic field ℚ _p . Let q be a unit of ℤ _p , q not a root of unity and V _q be the closure of the set {q ⁿ/n ∈ ℤ} and let

(V _q ,K) be the Banach algebra of the continuous functions from V _q to K. Let τ _q be the operator on

(V _q ,K) defined by τ _q (f)(x) = f(qx), f ∈

(V _q ,K). In her article [10] (see also [11]), A. Verdoodt constructs orthonormal bases associated to specific operators that commute with τ _q . Let q ∈ K such that |q − 1| < 1, q not a root of unity. Let

(ℤ _p ,K) be the Banach algebra of continuous functions from ℤ _p to K. We give here, as in umbral calculus, a bijective correspondence between a class of orthonormal bases of

(ℤ _p , K) and a class of linear continuous operators which commute with the translation operator τ ₁: τ ₁(f)(x) = f(x + 1).