B.A. Mikhailov

The theorem of readout long since known at us as the theorem of V.A. Kotelnikov's, abroad - as C. Shannon's theorem, and then everywhere - as theorem of Whittaker's, Kotelnikov's, Shannon's (WКSH theorem), is a basis of the decision of set of the theoretical and practical problems demanding digitization and exact interpolation of signals with finite by a spectrum. WКSH theorem and its generalisations speak about possibility of restoration of linear transformation of function with finite a spectrum on readout of its other linear transformation since the same width finite a spectrum. Occurrence of a high-precision method of the decision of a return problem by a reduction to the perfect part, the decision of accompanying problems, introduction of a quantitative measure infinite spectrum in the form of the relation of width of a spectrum to frequency of digitization have allowed to formulate the theorem about made (high-precision, with the superpermission) signal restoration on readout of its other linear transformation with infinity a spectrum. Before theorems of readout covered only the elementary return problems in which accuracy and stability of decisions were not a problem, and the superresolution was not the purpose. According to the actions ordered by the theorem, perfect restoration of an entrance signal is made in the constructive, easily realised form. For efficiency of the theorem important, that: - Analytical continuation of a spectrum dares replacement of narrow-band reactions by entrance basic signals finity basic signals, and actions exclusively over narrow-band signals, that is, without level increase fluctuation noise; - Moreover, integrated character of definition of factors Fourie lowers level narrow-band additive fluctuation noise and level of external noise; - Noise of imposing of spectra are excluded from the decision, that too is unique property of the theorem. As a result, as shows the made analysis, the effect of restraint of influence of noise with increase of level of the superresolution is observed. Examples and illustrations when without application of the offered theorem the return problem essentially does not dare are resulted. Article materials allow to conclude, that the theorem of readout for signals with infinity a spectrum expands a circle of solved problems, does real a superresolution problem, in some cases allows to refuse essentially problematic coordination of frequency pass-bands of a filter part and samplings part of system.