Distributional convolutors for Fourier transform
A distributional Fourier analysis was developed in a series of six papers by \textit{K.\,B.\thinspace Howell} [J.~Math.\ Anal.\ Appl.\ 168, No.\,2, 342--350 (1992; Zbl 0776.42012); ibid.\ 173, No.\,2, 419--429 (1993; Zbl 0776.42013); ibid.\ 175, No.\,1, 257--267 (1993; Zbl 0776.42014); ibid.\ 180, No.\,1, 79--92 (1993; Zbl 0811.42004); ibid.\ 187, No.\,2, 567--582 (1994; Zbl 0819.46028); ibid.\ 193, No.\,3, 832--838 (1995; Zbl 0869.46017)]. The work done by the authors in the present paper complements this work, i.e., it fills up some gaps in the work done by Howell. This is exemplified by their Theorem 3.1, amongst many other theorems stated and proved; it states the following. Let \(m\in \mathbb Z\), \(m<0\), and \(T\in {\mathcal O}'_{c,G,m}\). Then \[ T(x) = \lim_{r\to\infty} \int^{+r}_{-r} F(T)(z)e^{-ixz}\,dz \] in the sense of convergence in \(G'\). Here, \(G\) and \({\mathcal O}_{c,G,m}\), are Fréchet spaces and \({\mathcal O}'_{c,G,m}\) and \(G'\) are their corresponding dual spaces, respectively.
scientific article; zbMATH DE number 5078938