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Extraction of Several Harmonics from Trigonometric Polynomials. Fejer-Type Inequalities

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Abstract

Given a trigonometric polynomial \({T_n}(t) = \sum\nolimits_{k = 1}^n {{\tau _k}\left( t \right),{\tau _k}\left( t \right): = {a_k}\cos kt + {b_k}\sin kt}\) we consider the problem of extracting the sum of harmonics \(\sum \tau_{\mu_s}(t)\) prescribed orders µs by the method of amplitude and phase transformations. Such transformations map the polynomials Tn(t) into similar ones using two simple operations: the multiplication by a real constant X and the shift by a real phase λ, i.e., Tn(t) → XTn(t — λ). We represent the sum of harmonics as a sum of such polynomials and then use this representation to obtain sharp Fejer-type estimates.

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Funding

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (state assignment 1.574.2016/1.4) and by the Russian Foundation for Basic Research (project no. 18-01-00744).

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Authors and Affiliations

  1. Vladimir State University Named after Alexander and Nikolay Stoletovs, ul. Gor’kogo 87, Vladimir, 600000, Russia

    D. G. Vasilchenkova & V. I. Danchenko

Authors
  1. D. G. Vasilchenkova
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  2. V. I. Danchenko
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Corresponding authors

Correspondence to D. G. Vasilchenkova or V. I. Danchenko.

Additional information

Russian Text © The Author (s), 2020, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2020, Vol. 308, pp. 101–115.

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Vasilchenkova, D.G., Danchenko, V.I. Extraction of Several Harmonics from Trigonometric Polynomials. Fejer-Type Inequalities. Proc. Steklov Inst. Math. 308, 92–106 (2020). https://doi.org/10.1134/S0081543820010083

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