A Paley-Wiener theorem for the Hankel transform of Colombeau type generalized functions
The discovery of generalized functions is one of the most distinguished and interesting fields, but Colombeau has developed a new theory of generalized functions by generalizing onto the transformations. The properties, such as, convolution, convolutors, tempered generalized functions of the Hankel transform for Colombeau type of generalized function were introduced in the recent years. In the present paper, after defining some of the above properties in a systematic manner, the authors derive a Paley-Wiener theorem for the Hankel transformation \[ h_\mu(\varphi)(y)= \int^\infty_0 (xy)^{1/2} J_\mu(xy)\varphi(x) dx,\quad y\in(0,\infty), \] where \(J_\mu\) denotes the Bessel function of the first kind and of order \(\mu>-1/2\) on generalized function of the Colombeau type spaces \({\mathcal H}_{\mu,\tau,h}\) and \({\mathcal H}_{\mu,\tau,c}\), which are subspaces of \({\mathcal H}_{\mu,\tau}\). For a Paley-Wiener theorem, few lemmas have been proved and few properties for the Colombeau type spaces are justified for the Hankel transformation. The main theme and importance of this paper is, that, a Paley-Wiener type theorem is established for the Hankel transform in the setting of the tempered generalized functions of Colombeau type.
tempered generalized functions of Colombeau type
scientific article; zbMATH DE number 1271510