V.F. Kravchenko, D.V.Churikov

For the first time a proof of existence of the inverse Fourier transform (IFT) in L1 and L2 on the basis of the theory of atomic functions (AF) are substantiate. for this purpose subintegral function is modified and according to Fubini theorem that double integral is reduced to the iterated integral. The IFT is proved by using AF in L1 in two ways, and extended to the L2. The modified Kravchenko atomic transformations (AT) are constructed and justified. With them were investigated The model signals that can can be viewed as narrow-band, broadband or ultra wide band were analyzed with help of AT. Modified atomic transformations allows better time and frequency resolution. The new complex analytic wavelets Kravchenko are obtained. The advantage of the WA-systems of Kravchenko functions in comparison with Morlet wavelet are the exact fulfillment of the condition of zero mean for any value of the modulation parameter. Even if it is less than the maximum frequency of mother AF. In such a case the distortion of the real part of wavelet spectrum is observed. As the mother AF is finite then while scaling argument type of does not change. The quality functional, which allows to select the most suitable wavelet bases for specific tasks is considered. Complex Kravchenko wavelets applied to time series analysis in astrophysics (Apparent magnitude of quasar 3C273 and the Wolf numbers changes). The numerical experiments and physical analysis of the results confirm the effectiveness of the proposed changes, as well as new constructions of wavelets in the problems of digital signal and image processing.