A formula for the Fourier transform of certain odd differentiable functions.
type: Item
instance of: scholarly article
title: A formula for the Fourier transform of certain odd differentiable functions.
zbMATH Open document ID: 1051.42006
DOI: 10.1016/S0022-247X(03)00243-9
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-10-14T00:00:00Z
review text: The author proves a Fourier transform formula for a certain class of differentiable functions \(K(x)=B(x)/x\) defined on the real line, relating the global behaviour of the Fourier transform of \(K\) to the function \(B\). Let \(\widehat{f}(z)=\int{f(x)e^{-izx} \,dx}\), \(z\) real, denote the Fourier transform of \(f\), and let \(A\approx B\) mean the existence of positive constants \(C_1\) and \(C_2\) such that \(C_1B\leq A\leq C_2B\). The following theorem is proved. Theorem: Let \(K\) be positive for \(x>0\) and odd. Suppose there exist constants \(c_1\) and \(c_2\), \(0< c_1\leq c_2<2\), such that \[ c_1\frac{K(x)}{x}\leq -K'(x)\leq c_2\frac{K(x)}{x} \] for \(x> 0\). Then, \(\widehat{K}\) exists as a tempered distribution and can be identified with a function (which we also call \(\widehat{K}\)) such that for \(z>0\), \[ i\widehat{K}(z)\approx B(\pi/z). \]
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 26A12
zbMATH DE Number: 1992550
zbMATH Keywords: Fourier transform
zbMATH Keywords: tempered distribution
zbMATH Keywords: slowly varying function
zbMATH Keywords: asymptotic behaviour
author: Ryan M. Berndt
reviewed by: B. P. Duggal
MaRDI profile type: Publication
cites work: Q4726487
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description: scientific article; zbMATH DE number 1992550