Fourier-Based Spectral Analysis with Adaptive Resolution

Khilko, Andrey
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Despite being the most popular methods of data analysis, Fourier-based techniques suffer from the problem of static resolution that is currently believed to be a fundamental limitation of the Fourier Transform. Although alternative solutions overcome this limitation, none provide the simplicity, versatility, and convenience of the Fourier analysis. The lack of convenience often prevents these alternatives from replacing classical spectral methods - even in applications that suffer from the limitation of static resolution. This work demonstrates that, contrary to the generally accepted belief, the Fourier Transform can be generalized to the case of adaptive resolution. The generalized transform provides backward compatibility with classical spectral techniques and introduces minimal computational overhead.
Comment: 31 pages including 10 figures. Fixed problem with computational complexity, so section 10 is rewritten and 3 more figures are added. Also, a way of optimal computation is suggested. Corrected a couple of typos in formulas. Several language and style corrections. Two additional references are added
Physics - Data Analysis, Statistics and Probability, Mathematics - General Mathematics, Mathematics - Numerical Analysis