Abel-Poisson Partial Sums in Signal Theory

The mathematical component of modern signal processing increasingly requires the use of approximation theory methods to obtain the necessary information from the investigated signal. The special value of the application of approximation theory methods is observed in digital signal processing, namely, in problems related to image recognition, the selection of useful information taking into account background noise, etc. In this work, the method of Abel-Poisson partial sums has been introduced as one of the methods of Fourier series summation. The order of saturation of this summation method has been established, as well as the classes of functions on which this saturation order is achieved. The introduced Abel-Poisson partial sums have been presented in the form of the corresponding integral. Abel-Poisson interpolation polynomials have been constructed using the latter. The interpolation polynomials obtained in the work have been compared with the interpolation polynomials used in signal theory, constructed using the Fejer method. In the course of the research, we have established that the introduced Abel-Poisson interpolation polynomials are more predictable when solving the applied problems.