The Mathematica Journal
Volume 9, Issue 2

Search

In This Issue
Articles
Tricks of the Trade
In and Out
Trott's Corner
New Products
New Publications
Calendar
News Bulletins
New Resources
Classifieds

Download This Issue 

About the Journal
Editorial Policy
Staff
Submissions
Subscriptions
Advertising
Back Issues
Contact Information

p-adic Arithmetic
Stany De Smedt

Function Expansion

For the space of all continuous functions from to (we can simply transfer the definitions from classical analysis to non-Archimedean analysis by adapting the valuations), there exist two well-known bases. On the one hand, we have the Mahler basis, consisting of the polynomials . On the other hand, there is the Vanderput basis, consisting of locally constant functions , where and for , is the characteristic function of the open ball with center and radius . Each continuous function can then be written as

Here is defined as follows. For every , there exists a p-adic expansion with . Then .

In this section of the package we implement these expansions as Mahler[f,x,n] and Vanderput[f,x,n,p], which calculate the first coefficients of the function according to the respective basis. Note that the Vanderput expansion depends on , while the Mahler expansion does not.

Here are some examples.



     
About Mathematica | Download Mathematica Player
Copyright © Wolfram Media, Inc. All rights reserved.