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A note on some expansions of p-adic functions

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by (φm)m∈N0 . The system (φm)m∈N0 is defined in the space C(Zp, Cp) of Cp-valued continuous functions on Zp. J. Rutkowski has also considered some questions concerning expansions of functions from C(Zp, Cp) with respect to (φm)m∈N0 . This paper is a remark to Rutkowski’s paper. We define another system (hn)n∈N0 in C(Zp, Cp), investigate its properties and compare it to the system defined by Rutkowski. The system (hn)n∈N0 can be viewed as a p-adic analogue of the well-known Haar system of real functions (see [1]). It turns out that in general functions are expanded much easier with respect to (hn)n∈N0 than to (φm)m∈N0 . Moreover, a function in C(Zp, Cp) has an expansion with respect to (hn)n∈N0 if it has an expansion with respect to (φm)m∈N0 . At the end of this paper an example is given of a function which has an expansion with respect to (hn)n∈N0 but not with respect to (φm)m∈N0 . Throughout the paper the ring of p-adic integers, the field of p-adic numbers and the completion of its algebraic closure will be denoted by Zp, Qp and Cp respectively (p prime). In addition, we write N0 = N ∪ {0} and E = {0, 1, . . . , p− 1}. The author would like to thank Jerzy Rutkowski for fruitful comments and remarks that permitted an improvement of the presentation.