Geodesic
$p$-Adic Fourier Theory of Differentiable Functions
Documenta mathematica, Tome 23 (2018), pp. 939-967
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Let K be a finite extension of Qp​ of degree d and OK​ its ring of integers; let Cp​ be the completed algebraic closure of Qp​. The Fourier polynomials Pn​:OK​→Cp​ show that the topological algebra of all locally analytic distributions μ:Cla(OK​,Cp​)→Cp​ is, by μ↦∑μ(Pn​)Xn, isomorphic to that of all power series in Cp​[[X]] that converge on the open unit disc of Cp​. Given a real number r≥d, we determine the power series that correspond under this isomorphism to all distributions μ:Cr(OK​,Cp​)→Cp​ that extend to all r-times differentiable functions (as arisen in the p-adic Langlands program): A function f:OK​→Cp​ is r-times differentiable if and only if f(x)=Σan​Pn​(x) with ∣an​∣nr/d→0 as n→∞.
DOI : 10.4171/dm/639
Classification : 11S31, 11S80, 12J25, 14G22, 32P05, 46S10
Mots-clés : Fourier transform, Mahler basis, Amice transform, Lubin-Tate formal group, Taylor polynomials 
@article{10_4171_dm_639,
     author = {Enno Nagel},
     title = {$p${-Adic} {Fourier} {Theory} of {Differentiable} {Functions}},
     journal = {Documenta mathematica},
     pages = {939--967},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/639},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/639/}
}
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Enno Nagel. $p$-Adic Fourier Theory of Differentiable Functions. Documenta mathematica, Tome 23 (2018), pp. 939-967. doi: 10.4171/dm/639

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