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Volume 9, Issue 2


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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.

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