M.P. Slichenko

JSC «Concern «Sozvezdie» (Voronezh, Russia)

Periodic functions describe phenomena of various nature and are used in various fields of science and technology. In practice, the problem of restoring such periodic functions from their discrete non-equidistant samples is widespread, especially when processing the results of measurements of such functions. However, the existing approach to interpolation by non-equidistant Kotelnikov series leads to errors due to the truncation of the infinite series. This error can be eliminated and the accuracy of interpolation can be improved by modifying the Kotelnikov series based on the periodicity of the original function. Along with the analytical representation of the original function and its derivative, in many scientific problems it is required to calculate the integral of a function given by a discrete set of samples.

Theorems on the representation of the integral of a periodic function with a finite Fourier spectrum as a finite sum for functions of one and several variables are formulated in this paper. Based on the modified Kotelnikov series, theorems are formulated on the representation of the integral of a periodic function with a finite Fourier spectrum as a finite sum. In contrast to the use of an interpolation series with a kernel in the form of spectra of atomic functions, such a representation does not require oversampling of the original function, and also makes it possible to increase the integration accuracy by zeroing the truncation error.

The results are generalized to the general case of integrating a complex function of several variables, when the integrand is factorized with a periodic factor having a finite spectrum.

The formulated theorems can find practical application in many fields of science and technology, including radiophysics, electrodynamics, antenna theory, oscillation theory, optics, holography and radio astronomy, in solving problems of mathematical physics, digital signal and image processing. The application of these theorems in electrodynamics and antenna theory will improve the accuracy of calculating the characteristics of the electromagnetic radiation field based on the measurement results. In particular, the theorems make it possible to express the directivity of an antenna in terms of readings of its radiation pattern. The results have a rather high potential for application in modern computer-aided design systems in the numerical solution of electrodynamic simulation
problems.

Slichenko M.P. Theorems on the representation of the integral of a periodic function with finite Fourier spectrum as a finite sum.
Radiotekhnika. 2023. V. 87. № 5. P. 134−142. DOI: https://doi.org/10.18127/j00338486-202305-14 (In Russian)

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