\(p\)-adic Fourier theory of differentiable functions
type: Item
instance of: scholarly article
title: \(p\)-adic Fourier theory of differentiable functions
zbMATH Open document ID: 1407.11135
author: Enno Nagel
published in: Documenta Mathematica
publication date: 2018-08-10T00:00:00Z
review text: Let \(\mathbb{Q}_p\) be the field of \(p\)-adic numbers and let \(\mathbb{C}_p\) be the completion of an algebraic closure of \(\mathbb{Q}_p\). Let \(K/\mathbb{Q}_p\) be a finite subextension of \(\mathbb{C}_p/\mathbb{Q}_p\) of degree \(d\) and let \(\mathcal{G}(X,Y)\) be a Lubin-Tate formal group law for \(K\). Let \(\log_{\mathcal{G}}(X)\in KX\) be the logarithm of \(\mathcal{G}\) and let \(\exp(X)\in\mathbb{Q}X\) be the usual exponential series. Then there are polynomials \(P_n(Y)\in K[Y]\) such that \[ \exp(Y\log_{\mathcal{G}}(Z))=\sum_{n=0}^{\infty} P_n(Y)Z^n. \] Say that \(P_n(Y)\) is the Fourier polynomial of degree \(n\). Let \(r\) be a real number such that \(r\geq d\). There is a notion of \(r\) times differentiability for functions \(f:\mathcal{O}_K\rightarrow\mathbb{C}_p\). The main theorem of this paper states that \(f\) is \(r\) times differentiable if and only if there exist \(a_n\in\mathbb{C}_p\) such that \(f(x)=\sum_{n=0}^{\infty}a_nP_n(x)\) and \(\lim_{n\rightarrow\infty}|a_n|n^{r/d}=0\). This result had been known previously in the case \(K=\mathbb{Q}_p\).
reviewed by: Kevin Keating
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 32P05
Mathematics Subject Classification ID: 43A32
Mathematics Subject Classification ID: 46S10
zbMATH DE Number: 6915996
zbMATH Keywords: Fourier transform
zbMATH Keywords: Mahler basis
zbMATH Keywords: Amice transform
zbMATH Keywords: Lubin-Tate formal group
zbMATH Keywords: Taylor polynomials
MaRDI profile type: Publication
cites work: Interpolation $p$-adique
cites work: Q3579566
cites work: Multivariable Lubin-Tate (\(\varphi, \Gamma\))-modules and filtered \(\varphi-\)modules
cites work: Integral structures on \(p\)-adic Fourier theory
cites work: Le programme de Langlands $p$-adique
cites work: Spaces of functions of class \(C^r\) on \(\mathcal O_F\)
cites work: Q4177670
cites work: Formal complex multiplication in local fields
cites work: Q2996079
cites work: Q2990073
cites work: \(p\)-adic Taylor polynomials
cites work: Q3401901
cites work: Q3346515
cites work: Q2761412
cites work: \(p\)-adic Fourier theory
cites work: Banach space representations and Iwasawa theory
DOI: 10.25537/DM.2018V23.939-967
description: scientific article; zbMATH DE number 6915996