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How should one go about doing numerical analysis with $p$-adic numbers?

By that I mean, how should one go about implementing numerical integration (using analogues of Newton-Cotes or perhaps Gaussian quadrature formulae) and differentiation (using analogues of finite-differencing schemes, and perhaps a Richardson approach) of functions from $\mathbb{Q}_p^d$ to $\mathbb{R}$? Is there a theory of error estimates for such approximations?

More specifically:

  1. Is anything known about a systematic way of choosing $N$ points $\xi_i\in\Omega\subset\mathbb{Q}_p^d$ and weights $\omega_i\in\mathbb{R}$ such that $$ \int_{\Omega} f(x)\,{\rm d}_p^dx = \sum_{i=1}^N\omega_if(\xi_i) + {\rm O}(N^{-\alpha}) $$ for some $\alpha>0$, where ${\rm d}_p^qx$ is the Haar measure of $\mathbb{Q}_p^d$?

    I would suspect that one should be able to achieve $\alpha=1/2$ by picking $\xi_i$ randomly i.i.d. uniformly w.r.t. the Haar measure and putting $\omega_i=\frac{1}{N}$ as in naive Monte Carlo integration in $\mathbb{R}^d$. Is that so? And/or can one do better using a deterministic procedure?

  2. For numerical differentiation one can probably use an integral representation of the Vladimirov operator and apply a quadrature formula (cf. the preceding question). Is there any reason to expect complications with the error when doing so? In particular, how should one deal with the point $x=y$, where the kernel of the Vladimirov operator is very singular?

Note that I am specifically interested in real-valued functions.

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    $\begingroup$ I don't know about numerical analysis, but when it comes to algorithms for computing with $p$-adic numbers, and a detailed study of $p$-adic precision issues, Xavier Caruso is the expert. You should look at his webpage. $\endgroup$
    – Laurent Berger
    Mar 10, 2020 at 8:03
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    $\begingroup$ Thanks for the pointer to Xavier Caruso. I had already stumbled on some of this expository writings about implementing $p$-adic arithmetic and its precision issues, but as far as I could see that was all in the context of functions $\mathbb{Q}_p^d\to\mathbb{Q}_p$, not $\mathbb{Q}_p^d\to\mathbb{R}$. $\endgroup$
    – gmvh
    Mar 10, 2020 at 19:13
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    $\begingroup$ What condition do you have on $f$? Presumably it is not locally constant. $\endgroup$
    – Chris Wuthrich
    Mar 13, 2020 at 18:34
  • $\begingroup$ I'd suspect that the possibility of achieving a certain $\alpha$ may very well depend on the conditions $f$ satisfies. If it is locally constant, the error term ought to vanish at some point (unless the $\xi_i$ are chosen in a deliberately perverse way), so it won't be ${\rm O}(N^{-\alpha})$, is that what you mean? $\endgroup$
    – gmvh
    Mar 13, 2020 at 20:24
  • $\begingroup$ @gmvh can I ask for motivation behind this? $\endgroup$
    – bambi
    Mar 14, 2020 at 8:25

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