A \(p\)-adic analog of Wirtinger's inequality
type: Item
instance of: scholarly article
title: A \(p\)-adic analog of Wirtinger's inequality
zbMATH Open document ID: 0884.11051
author: Yanyan Li
published in: Michigan Mathematical Journal
publication date: 1996-10-13T00:00:00Z
review text: In classical Fourier analysis, for a nonvanishing complex-valued \(C^1\)-function \(f\) on the unit circle one proves that \(f\) has no zeros on some arc whose length can be expressed in terms of Fourier coefficients of \(f\) and \(f'\). In the present paper, the following \(p\)-adic version is proved. (Let \(\mathbb{Q}_p\) be the \(p\)-adic number field, let \(\mathbb{Z}_n\) be its valuation ring, and let \(\mathbb{C}_n\) be the completion of the algebraic closure of \(\mathbb{Q}_p\).) Let \(f:\mathbb{Z}_p\to K\) be nonzero, uniformly differentiable, where \(K\supset \mathbb{Q}_p\) is a tame extension of \(\mathbb{Q}_p\). Then \(f\) has no zeros on some coset of \(p^{M(f)}\mathbb{Z}_p\). Here \(M(f)\) is some well-defined constant expressed in terms of the Fourier transform \(\widehat f\) (which is defined on the `circle' \(\{\omega\in\mathbb{C}_p: \omega^{p^n}=1\) for some \(n\in\mathbb{N}\}\). A connection between \(M(f)\) and the behaviour of \(f'\) is established. Numerous examples are given.
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 26E30
Mathematics Subject Classification ID: 12J25
zbMATH DE Number: 934202
zbMATH Keywords: Wirtinger's inequality
zbMATH Keywords: \(p\)-adic Fourier transform
zbMATH Keywords: zero-free regions
reviewed by: Wim Schikhof
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005467
OpenAlex ID: W2082780918
DOI: 10.1307/MMJ/1029005467
description: scientific article; zbMATH DE number 934202