Fourier transforms and convolutions of generalized measures defined on the dyadic field
type: Item
instance of: scholarly article
title: Fourier transforms and convolutions of generalized measures defined on the dyadic field
zbMATH Open document ID: 0542.42016
DOI: 10.1007/BF01956772
author: N. E. Zubov
published in: Acta Mathematica Hungarica
publication date: 1983-01-01T00:00:00Z
review text: The Walsh Fourier transform of a dyadic measure m is defined as \(m{\hat{\;}}(I_ n(x))=(2^{-n})\int^{(2^ n)^-}_{0}w_ y(m)dx. w_ y\) is a Walsh function and \(I_ n(x)\) is the dyadic interval of rank n containing x. The convolution of two dyadic measures m and m' is defined as \[ (m*m')[p/2^ n,\quad p+1^-/2^ n]=\sum^{n}_{k=0}m[(p+k)/2^ n,\quad(p+k)+1^-/2^ n]m'[k/2^ n,\quad(k+1^-)/2^ n]. \] The main result of this paper is the following. Theorem: If dyadic measures m,m' satisfy \(\sum^{\infty}_{p=0}| m[p/2^ n,\quad(p+1^-);2^ n]|<\infty,\) and \(\sup_{x}| m'(I_ n(x))|<\infty\) for each \(n=0,\pm 1,...\), then for each dyadic interval I \[ (m*m'){\hat{\;}}I=\int_{I}\hat f(x)m{\hat{\;}}'(dx), \] \(\hat f\) is a locally integrable function satisfying m{\^{\ }}\(=m_{\hat f}\).
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 3861760
zbMATH Keywords: dyadic measure
zbMATH Keywords: quasi-measure
zbMATH Keywords: Walsh Fourier transform
MaRDI profile type: Publication
cites work: On the Walsh Functions
cites work: The Generalized Walsh Functions
full work available at URL: https://doi.org/10.1007/bf01956772
OpenAlex ID: W2057238689
description: scientific article; zbMATH DE number 3861760