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