Hartley transform for \(\mathcal L_p\) Boehmians and spaces of ultradistributions (Q2909873)

The paper attempts to extend the Hartley transform to the setting of Boehmians.NEWLINENEWLINE Reviewer's remarks:NEWLINENEWLINE1. The condition of convolution that (as title) appears in line 6 from below as property (iii) of the Hartley transform, exposes some unawareness of the theory of convolution of the Hartley transform. For a correct representation, the authors are advised to refer to [\textit{K. J. Oleiniczak}, ``The Hartley transform''. In: Poularikas, A. D. (ed.), The Transforms and Application Handbook, CRC Press, Boca Raton, FL, pp. 281--330 (2000)], where these conditions are completely discussed.NEWLINENEWLINE2. The problem that is discussed in the paper under review is partly a replica of the paper [\textit{D. Loonker, P. K. Banerji} and \textit{L. Debnath}, ``Hartley transform for integrable Boehmians'', Integral Transforms Spec. Funct. 21, No. 5--6, 459--464 (2010; Zbl 1208.46042)].NEWLINENEWLINE3. Needless to say that the concept of Boehmians depends on the convolution. In the absence of a correct definition of the convolution of the Hartley transform in the present paper, the proofs of theorems concerning the Hartley transform for integrable Boehmians are not proper and remain unexplained.NEWLINENEWLINE4. Theorem 5.4, page 442 of the paper under review is similar to a publication by \textit{D. H. Nair, D. Loonker} and \textit{P. K. Banerji} [``On ultradistributional Hartley transform'', The Aligarh Bull. Math. 29 (1), 47--53 (2010)]. Moreover, the proof of Theorem 5.5 is not complete.