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 ℚp. Let q be a unit of ℤp, q not a root of unity and Vq be the closure of the set {qn/n ∈ ℤ} and let Open image in new window (Vq,K) be the Banach algebra of the continuous functions from Vq to K. Let τq be the operator on Open image in new window (Vq,K) defined by τq(f)(x) = f(qx), f ∈ Open image in new window (Vq,K). In her article [10] (see also [11]), A. Verdoodt constructs orthonormal bases associated to specific operators that commute with τq. Let q ∈ K such that |q − 1| < 1, q not a root of unity. Let Open image in new window (ℤp,K) be the Banach algebra of continuous functions from ℤp to K. We give here, as in umbral calculus, a bijective correspondence between a class of orthonormal bases of Open image in new window (ℤp, K) and a class of linear continuous operators which commute with the translation operator τ1: τ1(f)(x) = f(x + 1).