Pointwise and directional regularity of nonharmonic Fourier series
type: Item
instance of: scholarly article
title: Pointwise and directional regularity of nonharmonic Fourier series
zbMATH Open document ID: 1193.42126
published in: Applied and Computational Harmonic Analysis
publication date: 2010-05-17T00:00:00Z
full work available at URL: PDRNFS.pdf
review text: The author studies nonharmonic Fourier series, i.e., functions of the form \(\sum_{n\in N}a_ne^{i \lambda_n\cdot x}\) and in particular how the regularity of the series is related to the spacing of the frequencies. The main tool is the Gabor-wavelet transform defined by \(\frac 1{a^d} \int_{ R^d} f(x)e^{-i \lambda\cdot x} \phi_\Omega\left(\frac{x-b}a\right)\, dx\) where \(\phi\) is a smooth function such that its Fourier transform is supported in the unit ball centered at \(0\) and \(\phi_\Omega(x)= \phi(\Omega x)\) where \(\Omega\) is a rotation in \(\mathbb R^d\). Applications to everywhere irregularity of solutions of certain partial differential equations (Schrödinger's equation, simply supported beams) are given. By extending the results to the anistorpic case the author derives directional irregularity criteria as well.
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42B99
zbMATH DE Number: 5708939
zbMATH Keywords: directional regularity
zbMATH Keywords: nonharmonic Fourier series
zbMATH Keywords: Hölder classes
zbMATH Keywords: wavelet transform
zbMATH Keywords: Gabor wavelets
author: Stéphane Jaffard
MaRDI profile type: Publication
OpenAlex ID: W2164550108
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DOI: 10.1016/J.ACHA.2010.02.002
description: scientific article; zbMATH DE number 5708939