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International Journal of Engineering Mathematics
Volume 2014 (2014), Article ID 686785, 10 pages
Research Article

Inversion of Fourier Transforms by Means of Scale-Frequency Series

Center for Research in Applied Mathematics & Statistics (CRAMS), AUL, Lebanon

Received 18 February 2014; Revised 10 April 2014; Accepted 10 April 2014; Published 27 May 2014

Academic Editor: J. A. Tenreiro Machado

Copyright © 2014 Nassar H. S. Haidar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We report on inversion of the Fourier transform when the frequency variable can be scaled in a variety of different ways that improve the resolution of certain parts of the frequency domain. The corresponding inverse Fourier transform is shown to exist in the form of two dual scale-frequency series. Upon discretization of the continuous scale factor, this Fourier transform series inverse becomes a certain nonharmonic double series, a discretized scale-frequency (DSF) series. The DSF series is also demonstrated, theoretically and practically, to be rate-optimizable with respect to its two free parameters, when it satisfies, as an entropy maximizer, a pertaining recursive nonlinear programming problem incorporating the entropy-based uncertainty principle.