Extraction of harmonics from trigonometric polynomials by phase-amplitude operators

Vasilchenkova, D.; Danchenko, V.

doi:10.1090/spmj/1645

Extraction of harmonics from trigonometric polynomials by phase-amplitude operators
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by D. G. Vasilchenkova and V. I. Danchenko
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 32 (2021), 215-232

DOI: https://doi.org/10.1090/spmj/1645

Published electronically: March 2, 2021

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Abstract:

The method of phase-amplitude transformations is used for extraction of harmonics $\tau _{\mu }$ of a given order $\mu$ from trigonometric polynomials \begin{equation*} T_n(t)=\sum _{k=1}^n\tau _k(t), \quad \tau _k(t)\coloneq a_k\cos kt+b_k\sin kt. \end{equation*} Such transformations take polynomials $T_n(t)$ to similar polynomials by using two simplest operations: multiplication by a real constant $X$ and shift by a real phase $\lambda$, i.e., $T_n(t)\to X\cdot T_n(t-\lambda )$. The harmonic $\tau _{\mu }$ is extracted by addition of similar polynomials: \begin{equation*} \tau _{\mu }(t)=\sum _{k=1}^{m}X_k\cdot T_n(t-\lambda _k),\quad m\le n, \end{equation*} where the $X_k$ and $\lambda _k$ are defined by explicit formulas. Similar formulas for harmonics are obtained on a fairly large class of convergent trigonometric series. This representation yields sharp estimates of Fejér type for harmonics and coefficients of the polynomial $T_n$.

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