Extraction of harmonics from trigonometric polynomials by phase-amplitude operators,St. Petersburg Mathematical Journal

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Extraction of harmonics from trigonometric polynomials by phase-amplitude operators
St. Petersburg Mathematical Journal ( IF 0.8 ) Pub Date : 2021-03-02 , DOI: 10.1090/spmj/1645
D. G. Vasilchenkova , V. I. Danchenko

Abstract:The method of phase-amplitude transformations is used for extraction of harmonics $\tau _{\mu }$ of a given order $\mu$ from trigonometric polynomials

$\displaystyle T_n(t)=\sum _{k=1}^n\tau _k(t), \quad \tau _k(t)\colonequals a_k\cos kt+b_k\sin kt.$

Such transformations take polynomials

to similar polynomials by using two simplest operations: multiplication by a real constant

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:

$\displaystyle \tau _{\mu }(t)=\sum _{k=1}^{m}X_k\cdot T_n(t-\lambda _k),\quad m\le n,$

where the

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