Geodesic
Application of the Fourier modulation analysis method to the problem of derivatives recovery
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 1, pp. 44-59

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In this work, formulas are obtained for finding higher derivatives of a function (dependence), expressed through its Fourier coefficients (harmonic amplitudes). These formulas were found by differentiating both a function with a harmonically modulated argument and its Fourier series. The expressions given make it possible to find higher derivatives of functions numerically with any accuracy by means of different methods. For example, numerical integration of the Euler-Fourier formulas may be used for harmonic amplitudes. Derivatives also can be found experimentally when studying a nonlinear physical process by digitally recording the harmonic amplitudes of the dependence under study under simultaneous static and harmonic influences. The problem of reconstructing derivatives from Fourier coefficients is posed and an analysis of its correctness is performed. Formulas for estimating recovery errors are defined and recommendations are given to reduce these errors. Examples are given with different smoothness of analytical functions and of functions used to explain experiments: 1) an analytical function used to explain the magnetic properties of superconductors, whose Fourier coefficients are determined numerically with the error of the computational algorithm in the Mathcad software environment; 2) current-voltage characteristic (CVC) of 2 back-to-back semiconductor diodes, whose harmonic amplitudes were determined experimentally with a given measurement error of the device. The resulting derivative of the current-voltage characteristic is compared with the derivative obtained using numerical differentiation formulas; 3) an analytical function whose derivative has a discontinuity of the first kind. Errors in measuring Fourier coefficients were added artificially using a random number generator.
Keywords: Taylor-Fourier series, higher derivative of a function, the problem of restoring derivatives, problem correctness, errors in restoring derivatives
Mots-clés : Fourier coefficients, harmonic amplitudes 
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     author = {N. D. Kuz'michev},
     title = {Application of the {Fourier} modulation analysis method to the problem of derivatives recovery},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {44--59},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2024},
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N. D. Kuz'michev. Application of the Fourier modulation analysis method to the problem of derivatives recovery. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 1, pp. 44-59. http://geodesic.mathdoc.fr/item/SVMO_2024_26_1_a3/