Some problems in the theory of approximation of functions on the group of \(p\)-adic numbers
type: Item
instance of: scholarly article
title: Some problems in the theory of approximation of functions on the group of \(p\)-adic numbers
zbMATH Open document ID: 1411.46053
DOI: 10.1134/S207004661802005X
published in: \(p\)-Adic Numbers, Ultrametric Analysis, and Applications
publication date: 2018-11-07T00:00:00Z
review text: On a locally compact group, such as the complex numbers \(\mathbb{C}\), the Haar measure (normalized to take value \(1\) on \(\mathbb{Z}_p\)) is at one's disposal. In particular, it permits integrating functions \(f : \mathbb{Q}_p \to \mathbb{C}\) on the \(p\)-adic numbers. This paper studies the function spaces \[ L^\rho(\mathbb{Q}_p) := \left\{ f : \mathbb{Q}_p \to \mathbb{C} : \int_{\mathbb{Q}_p} |f|^\rho < \infty\right \} \] for \(1 \leq \rho \leq \infty\). For example, -- for \(\rho = 1\), these are the functions such that \[ \int_{\mathbb{Q}_p} |f| < \infty, \] -- for \(\rho = \infty\), these are the uniformly bounded functions. When a function is not locally constant, the notion of a Lipschitz-function captures the intuition that it varies little locally: While for \(\rho = \infty\) the definition of a Lipschitz-function in \(L^\rho(\mathbb{Q}_p)\) of order \(\alpha\) is the usual one of uniform boundedness \(|f(x) - f(y)| \leq C \cdot |x-y|^\alpha\) for a constant \(C > 0\), it is generalized to \(0 < \rho < \infty\) by uniformly bounding the integrals \[ \| f - \tau_h f \|_\rho=\int |f - \tau_h f|^\rho \] for translation operators \(\tau_h : f \mapsto f(\cdot + h)\) with \(h \to 0\) \(p\)-adically: 1. More exactly, the modulus of continuity \[ \omega(n, \rho) := \sup \{\| f - \tau_h f \|_\rho : |h| \leq p^{-n} \} \] measures how far \(f\) is from local constancy. If there is a constant \(C > 0\) such that \(\omega(n, \rho) \leq C \cdot p^{-n \alpha}\) for all \(n\), then \(f\) is a Lipschitz-function of order \(\alpha\). Theorem 1.5 shows that a function in \(L^\rho(\mathbb{Q}_p)\) is Lipschitz if and only if can be uniformly approximated by locally constant functions with respect to \(\| \cdot \|_\rho\) and relates the order to the speed of convergence. 2. Theorems 1.7, respectively 1.8, give a condition on the parameters \(\alpha\), \(\rho\) and \(\beta\), \(\sigma\) for which the space of Lipschitz functions of order \(\alpha\) in \(L^\rho(\mathbb{Q}_p)\…
reviewed by: Enno Nagel
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6974393
zbMATH Keywords: \(p\)-adic harmonic analysis
zbMATH Keywords: approximation of functions
zbMATH Keywords: modulus of continuity
zbMATH Keywords: \(p\)-adic Fourier transform
zbMATH Keywords: group of \(p\)-adic numbers
zbMATH Keywords: Lipschitz condition
zbMATH Keywords: embedding theorems
Wikidata QID: Q114074758
author: Sergei S. Platonov
MaRDI profile type: Publication
cites work: Q4064526
cites work: Q3235904
cites work: Q4003475
cites work: An analogue of the Titchmarsh theorem for the Fourier transform on the group of \(p\)-adic numbers
cites work: Q5816815
cites work: Moduli of continuity of functions, defined on a zero-dimensional group
cites work: On the modulus of continuity with respect to functions defined on Vilenkin groups
cites work: Hausdorff operators on \(p\)-adic linear spaces and their properties in Hardy, \(BMO\), and Hölder spaces
cites work: Q5558071
cites work: Q3999467
cites work: Q3915432
cites work: Q3715607
cites work: Q3655827
full work available at URL: s207004661802005x
OpenAlex ID: W2802095103
description: scientific article; zbMATH DE number 6974393