Extraction of harmonics from trigonometric polynomials by phase-amplitude operators

The method of phase-amplitude transformations is used for extraction of harmonics $ \tau _{\mu }$ of a given order $ \mu $ from trigonometric polynomials $\displaystyle T_n(t)=\sum _{k=1}^n\tau _k(t), \tau _k(t):= a_k\cos kt+b_k\sin kt.$     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: $\displaystyle \tau _{\mu }(t)=\sum _{k=1}^{m}X_k\cdot T_n(t-\lambda _k), m\le n,$     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$.