Spectrum Reconstruction Operator

Abstract

AbstractThe paper considers a solution for recovery of spectral density informativity for a generalized class of non-integrable signals. The standard Fourier transform, with a limited set of harmonics, reproduces the spectra of non-integrable signals distorted by modulation by bursts of higher harmonics. An alternative solution to this problem is proposed by developing a corrective weighting approximation system for 2π periodic functions. The weights correct the partial sums of the Fourier series. The approximation forms a system of hyperplanes associated with linear envelopes of partial sums of the series. Each successive plane forms a factor-space of dimension 1 with the previous ones. Then, according to the basic information on functional spaces, for such a system of hyperplanes there exists a normalized linear functional defined on any hyperplane not passing through the origin of coordinates. Given the relationship of the functional with the factorized linear envelopes of partial sums of Fourier series, we obtain a fixed periodic function of the weight correction approximation of the analyzed 2π periodic function. An example of modeling and use of the periodic approximation correction weight sequence with a graphical demonstration of the result of spectrum reconstruction is given.KeywordsFunctional spacesPartial sumsQuotient spaceCosetsLinear functional