One of major aspects of a problem of restoration (reconstruction) of the internal structure of various objects and other information about their properties considers the reception of the information on distribution of some physical characteristic from their experimentally obtained spectrum of integral projective data received by various methods (emissive, transmissive -- by a physical principle; acoustical, optical, laser, radiological -- by the type of used radiation, etc.). The given branch of investigation is known as a reconstructive tomography. The majority of methods for restoration of the images from their integral projections in reconstructive tomography are based on a fundamental generalized projective theorem with use of Fourier transform. The main difficulty in practical processing of the complex Fourier-like transforms (Laplace, Mellin transforms, etc.) despite of their convenience in analytical calculations and algebra becomes the insufficient speed of data processing for reception of the dynamic distribution image of an investigated physical characteristic. In the present work we examined the possibilities of application in various areas of a reconstructive tomography of the real-domain integral transform offered by Ralph Vinton Lyon Hartley in 1942 for study of a spectra of electrosignals, lastly named in his honor by Hartley transform. Some examples of an effective applications and basic properties of Hartley transform are presented in works of Ronald N. Bracewell. From middle of the 1960-the years there were offered various fast algorithms of calculation of discrete Fourier transform (FFT) which characterized by some advantage in speed of data processing in comparison with discrete FT, but however owing to its complexity and asymmetry FFT concedes in speed of processing to fast algorithms based on Hartley transform.