A discrete Hartley transform based on Simpson's rule (Q2795242)

The Hartley transform is the integral transform that maps a real-valued function into a real-valued frequency function, kernel of which is a cas-function. The Simpson's quadrature rule is introduced, with the help of which the discrete Hartley transform and its inverse are defined. A relation between this transform and its transpose is established. Component wise properties, analogous to the classical discrete Hartley transform, are given. Further, based on the Simpson's rule, the discrete Hartley transform, the circular convolution property and the cross correlation property, respectively, are established in Sections 2 and 3. Related to the Hartley transform, one may refer to the papers by \textit{D. Loonker} et al. [Integral Transforms Spec. Funct. 21, No. 5--6, 459--464 (2010; Zbl 1208.46042)]; by \textit{P. K. Banerji} and \textit{D. Nair} [Bull. Calcutta Math. Soc. 102, 147--252 (2010)] and by \textit{P. K. Banerji}, \textit{D. Nair} and \textit{D. Loonker} [The Aligarh Bull. Math. 29, 47--53 (2010)]. These publications are the classical applications of the Hartley transform to distribution spaces.