In the present work, we propose an extrapolation method, developed on the basis of spectral analysis, digital filtering, and the principle of demodulation of a complex signal, for predicting the beginning of cycle 25 of solar activity. The Wolf number and other measured characteristics of solar activity have a very complex spectral composition. The Sun, by the nature of its radiation, contributes a significant stochastic component to the observational data. The experimental data are known only up to the present, and the prediction is about bridging the gap in our data set. Mathematically, the prediction problem boils down to extrapolation of discontinuous functions, which leads to a Gibbs phenomenon that occurs at the point of discontinuity and makes prediction into the future impossible. To overcome this discontinuity, additional physical models describing a continuous process are most often used. This paper uses only the Wolf series of numbers from 1818 to 2020. The authors developed an original forecasting technique using Fourier series, digital filtering and representation of the complex process as modulated and subsequent demodulation. As a result of decomposing the complex signal by Fourier series into separate components, the spectral ranges characteristic of the Wolf number were singled out. Taylor's series was used for construction of prediction or extrapolation algorithms. The extraction of spectral ranges, characteristic for the investigated process, is carried out by means of sequential digital filtering methods and information compression in accordance with the cut-off frequency of the digital filter. For example, when selecting eleven-year cycles of solar activity, we have to compress the information by a factor of 160. With such a processing scheme, the forecasting starts with the ultralow-frequency component with a period of more than 11 years, successively moving to the ranges of higher frequencies. The use of spectral analysis and Chebyshev filtering showed the possibility to predict the low-frequency component for the full cycle period. The eleven-year component forecast obtained by the authors is in good agreement with the data of the Brussels Royal Center.
An inequality of Kostka numbers and Galois groups of Schubert problems