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padic Arithmetic
Function Expansion For the space of all continuous functions from to (we can simply transfer the definitions from classical analysis to nonArchimedean analysis by adapting the valuations), there exist two wellknown 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 padic 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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